Research Article On the Theory of Multilinear Singular ...downloads.hindawi.com/journals/jfs/2016/4149314.pdf · some classical singular integrals in harmonic analysis on them, such

Research ArticleOn the Theory of Multilinear Singular Operators with RoughKernels on the Weighted Morrey Spaces

Sha He and Xiangxing Tao

Department of Mathematics Zhejiang University of Science and Technology Hangzhou 310023 China

Correspondence should be addressed to Xiangxing Tao xxtaozusteducn

Received 27 February 2016 Accepted 10 July 2016

Academic Editor Shijun Zheng

Copyright copy 2016 S He and X Tao This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We study some multilinear operators with rough kernels For the multilinear fractional integral operators 119879119860Ω120572

and the multilinearfractional maximal integral operators119872119860

Ω120572 we obtain their boundedness on weighted Morrey spaces with two weights 119871119901120581(119906 V)

when119863120574119860 isin Λ120573(|120574| = 119898minus1) or119863120574119860 isin BMO (|120574| = 119898minus1) For the multilinear singular integral operators 119879119860

Ωand the multilinear

maximal singular integral operators 119872119860

Ω we show they are bounded on weighted Morrey spaces with two weights 119871119901120581(119906 V) if

119863120574119860 isin Λ

120573(|120574| = 119898 minus 1) and bounded on weighted Morrey spaces with one weight 119871119901120581(119908) if 119863120574119860 isin BMO (|120574| = 119898 minus 1) for

119898 = 1 2

1 Introduction and Main Results

Let us consider the following multilinear fractional integraloperator

119879119860

Ω120572119891 (119909) = int

R119899

Ω(119909 minus 119910)

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572+119898minus1119877119898(119860 119909 119910) 119891 (119910) 119889119910

0 lt 120572 lt 119899

(1)

and the multilinear fractional maximal operator

119872119860

Ω120572119891 (119909) = sup

119903gt0

1

119903119899minus120572+119898minus1

sdot int|119909minus119910|lt119903

1003816100381610038161003816Ω (119909 minus 119910) 119877119898 (119860 119909 119910) 119891 (119910)1003816100381610038161003816 119889119910

0 lt 120572 lt 119899

(2)

whereΩ isin 119871119904(119878119899minus1) (119904 gt 1) is homogeneous of degree zero inR119899119860 is a function defined onR119899 and119877

119898(119860 119909 119910)denotes the

119898th order Taylor series remainder of 119860 at 119909 expanded about119910 that is

119877119898(119860 119909 119910) = 119860 (119909) minus sum

|120574|lt119898

1

120574119863120574119860 (119910) (119909 minus 119910)

120574 (3)

120574 = (1205741 120574

119899) each 120574

119894 119894 = 1 119899 is a nonnegative integer

|120574| = sum119899

119894=1120574119894 120574 = 120574

1 sdot sdot sdot 120574

119899 119909120574 = 1199091205741

1sdot sdot sdot 119909

120574119899

119899 and 119863120574 = 120597|120574|

12059712057411199091sdot sdot sdot 120597

120574119899119909119899

We notice that if 120572 = 0 the above two operators 119879119860Ω120572

119872119860

Ω120572are the multilinear singular integral operator 119879119860

Ωand

themultilinearmaximal singular integral operator119872119860

Ωwhose

definitions are given as follows respectively

119879119860

Ω119891 (119909) = int

R119899

Ω(119909 minus 119910)

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899+119898minus1119877119898(119860 119909 119910) 119891 (119910) 119889119910 (4)

119872119860

Ω119891 (119909) = sup

119903gt0

1

119903119899+119898minus1


1003816100381610038161003816Ω (119909 minus 119910) 119877119898 (119860 119909 119910) 119891 (119910)1003816100381610038161003816 119889119910

(5)

For119898 = 1119879119860Ω120572

is obviously the commutator [119860 119879Ω120572] of119879

Ω120572

and 119860 [119860 119879Ω120572]119891(119909) = 119860(119909)119879

Ω120572119891(119909) minus 119879

Ω120572(119860119891)(119909) where

119879Ω120572

is the fractional integral operator given by

119879Ω120572119891 (119909) = int

R119899

Ω(119909 minus 119910)

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572119891 (119910) 119889119910 0 lt 120572 lt 119899 (6)

There are numerous works on the study of multilinearoperators with rough kernels If 119863120574119860 isin BMO (|120574| = 119898 minus 1)

Hindawi Publishing CorporationJournal of Function SpacesVolume 2016 Article ID 4149314 13 pageshttpdxdoiorg10115520164149314

2 Journal of Function Spaces

the 119871119901 boundedness of119879119860Ωwas obtained bymeans of a good-120582

inequality by Cohen and Gosselin [1] In 1994 Hofmann [2]proved that 119879119860

Ωis a bounded operator on 119871119901(119908) when nabla119860 isin

119861119872119874 and 119908 isin 119860119901 Recently Lu et al [3] proved 119879119860

Ωand119872119860

Ω

are bounded from 119871119901 to 119871119902 (1119901 minus 1119902 = 120573119899) when 119863120574119860 isin

Λ120573(|120574| = 119898 minus 1) while for multilinear fractional integral

operators Ding and Lu [4] showed the (119871119901(119908119901) 119871119902(119908119902))boundedness of 1198791198601 119860119896

Ω120572and119872119860

1119860119896

Ω120572(their definitions will

be given later) if 119863120574119860119895isin BMO (|120574| = 119898 minus 1) 119895 = 1 119896

After that Lu and Zhang [5] proved 119879119860Ω120572

is a boundedoperator from 119871119901 to 119871119902 (1119901 minus 1119902 = (120572 + 120573)119899) when119863120574119860 isinΛ120573(|120574| = 119898 minus 1)On the other hand the classical Morrey spaces were first

introduced by Morrey [6] to study the local behavior of solu-tions to second-order elliptic partial differential equationsFrom then on a lot of works concerning Morrey spacesand some related spaces have been done see [7ndash9] and thereferences therein for details In 2009 Komori and Shirai[10] first studied the weightedMorrey spaces and investigatedsome classical singular integrals in harmonic analysis onthem such as the Hardy-Littlewood maximal operator theCalderon-Zygmund operator the fractional integral opera-tor and the fractional maximal operator Recently Wang [11]discussed the boundedness of the classical singular operatorswith rough kernels on the weighted Morrey spaces

We note that many works concerning 119879119860Ω120572

119872119860

Ω120572 119879119860Ω

and119872119860

Ωhave been done on 119871119901 spaces or weighted 119871119901 spaces

when 119863120574119860 belongs to some function spaces for |120574| = 119898 minus1 However there is not any study about these operatorson weighted Morrey spaces Therefore it is natural to askwhether they are bounded on weighted Morrey spaces Theaim of this paper is to investigate the boundedness of 119879119860

Ω120572

119872119860

Ω120572 119879119860Ω and 119872119860

Ωon weighted Morrey spaces if 119863120574119860 isin

Λ120573(|120574| = 119898 minus 1) or 119863120574119860 isin BMO (|120574| = 119898 minus 1)

When 119863120574119860 isin Λ120573(|120574| = 119898 minus 1) we show 119879

119860

Ω120572and

119879119860

Ωare controlled pointwisely by the fractional singular

integral operators 119879Ω120572+120573

and 119879Ω120573

(their definition will begiven later) respectively Thus the problem of studying theboundedness of119879119860

Ω120572and119879119860

ΩonweightedMorrey spaces with

two weights could be reduced to that of 119879Ω120572+120573

and 119879Ω120573

When 119863120574119860 isin BMO (|120574| = 119898 minus 1) the boundedness of119879119860

Ω120572on weighted Morrey spaces with two weights is proved

by standard method However we could only obtain theboundedness of 119879119860

Ωon weighted Morrey spaces with one

weight for 119898 = 1 and 119898 = 2 since we need the 119871119901(119908)boundedness of 119879119860

Ωin our proof but to the best of our

knowledge there is not such bounds hold for119879119860Ωwhen119898 ge 3

For119872119860

Ω120572and119872119860

Ω we show they are controlled pointwisely by

119879119860

Ω120572and 119879119860

Ω respectively Thus it is easy to obtain the same

results for119872119860

Ω120572and119872119860

Ωas those of 119879119860

Ω120572and 119879119860

Ω

Before stating our main results we introduce somedefinitions and notations at first

A weight is a locally integrable function on R119899 whichtakes values in (0infin) almost everywhere For a weight 119908

and a measurable set 119864 we define 119908(119864) = int119864119908(119909)119889119909 the

Lebesgue measure of 119864 by |119864| and the characteristic functionof 119864 by 120594

119864 The weighted Lebesgue spaces with respect to the

measure 119908(119909)119889119909 are denoted by 119871119901(119908) with 0 lt 119901 lt infinWe say a weight 119908 satisfies the doubling condition if thereexists a constant 119863 gt 0 such that for any ball 119861 we have119908(2119861) le 119863119908(119861) When 119908 satisfies this condition we denote119908 isin Δ

2for short

Throughout this paper 119861(1199090 119903) denotes a ball centered at

1199090with radius 119903 Let 119876 be a cube with sides parallel to the

axes For119870 gt 0119870119876 denotes the cube with the same center as119876 and side-length being119870 times longer When 120572 = 0 we willdenote 119879

Ω120572 119879119860Ω120572

119872119860

Ω120572by 119879

Ω 119879119860Ω119872119860

Ω respectively And for

any number 119886 1198861015840 stands for the conjugate of 119886 The letter 119862denotes a positive constant that may vary at each occurrencebut is independent of the essential variable

Next we give the definition of weighted Morrey spaceintroduced in [10]

Definition 1 Let 1 le 119901 lt infin let 0 lt 120581 lt 1 and let 119908 be aweight Then the weighted Morrey space is defined by

119871119901120581(119908) fl 119891 isin 119871119901loc (119908)

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908) lt infin (7)

where

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908) = sup

119861

(1

119908 (119861)120581int119861

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119901119908 (119909) 119889119909)

1119901

(8)

and the supremum is taken over all balls 119861 in R119899

When we investigate the boundedness of the multilinearfractional integral operator we need to consider the weightedMorrey space with two weights It is defined as follows

Definition 2 Let 1 le 119901 lt infin let 0 lt 120581 lt 1 and let 119906 V be twoweights The two weights weighted Morrey space is definedby

119871119901120581(119906 V) fl 119891 1003817100381710038171003817119891

1003817100381710038171003817119871119901120581(119906V) lt infin (9)

where

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119906V) = sup

119861

(1

V (119861)120581int119861

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119901119906 (119909) 119889119909)

1119901

(10)

and the supremum is taken over all balls 119861 in R119899 If 119906 = Vthen we denote 119871119901120581(119906) for short

As is pointed out in [10] we could also define theweightedMorrey spaces with cubes instead of balls So we shall usethese two definitions of weighted Morrey spaces appropriateto calculation

Then we give the definitions of Lipschitz space and 119861119872119874space

Definition 3 The Lipschitz space of order 120573 0 lt 120573 lt 1 isdefined by

Λ120573(R119899) = 119891

1003816100381610038161003816119891 (119909) minus 119891 (119910)1003816100381610038161003816 le 119862

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

120573 (11)

and the smallest constant 119862 gt 0 is the Lipschitz norm sdot Λ120573

Journal of Function Spaces 3

Definition 4 A locally integrable function 119887 is said to be inBMO(R119899) if

119887lowast = 119887BMO = sup119861

1

|119861|int119861

1003816100381610038161003816119887 (119909) minus 1198871198611003816100381610038161003816 119889119909 lt infin (12)

where

119887119861=1

|119861|int119861

119887 (119910) 119889119910 (13)


At last we give the definition of two weight classes

Definition 5 A weight function 119908 is in the Muckenhouptclass 119860

119901with 1 lt 119901 lt infin if there exists 119862 gt 1 such that

for any ball 119861

(1

|119861|int119861

119908 (119909) 119889119909)(1

|119861|int119861

119908 (119909)minus1(119901minus1)

119889119909)

119901minus1

le 119862 (14)

We define 119860infin= ⋃

1lt119901ltinfin119860119901

When 119901 = 1 we define 119908 isin 1198601if there exists 119862 gt 1 such

that for almost every 119909

119872119908(119909) le 119862119908 (119909) (15)

Definition 6 A weight function 119908 belongs to 119860(119901 119902) for 1 lt119901 lt 119902 lt infin if there exists 119862 gt 1 such that such that for anyball 119861

(1

|119861|int119861

119908 (119909)119902119889119909)

1119902

sdot (1

|119861|int119861

119908 (119909)minus119901(119901minus1)

119889119909)

(119901minus1)119901

le 119862

(16)

When 119901 = 1 then we define 119908 isin 119860(1 119902) with 1 lt 119902 lt infin ifthere exists 119862 gt 1 such that

(1

|119861|int119861

119908 (119909)119902119889119909)

1119902

(ess sup119909isin119861

1

119908 (119909)) le 119862 (17)

Remark 7 (see [10]) If 119908 isin 119860(119901 119902) with 1 lt 119901 lt 119902 then

(a) 119908119902 119908minus1199011015840

119908minus1199021015840

isin Δ2

(b) 119908minus1199011015840

isin 1198601199051015840 with 119905 = 1 + 1199021199011015840

Now we state the main results of this paper

Theorem 8 If 0 lt 120572 + 120573 lt 119899 Ω isin 119871119904(119878119899minus1) (119904 gt 1)

is homogeneous of degree zero 1 lt 1199041015840lt 119901 lt 119899(120572 + 120573)

1119902 = 1119901 minus (120572 + 120573)119899 0 lt 120581 lt 119901119902 1199081199041015840

isin 119860(1199011199041015840 119902119904

1015840)

119863120574119860 isin Λ

120573(|120574| = 119898 minus 1) then

10038171003817100381710038171003817119879119860

Ω12057211989110038171003817100381710038171003817119871119902120581119902119901(119908119902)

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) (18)

10038171003817100381710038171003817119872119860

Ω12057211989110038171003817100381710038171003817119871119902120581119902119901(119908119902)

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) (19)

Theorem 9 If 0 lt 120573 lt 1 Ω isin 119871119904(119878119899minus1) (119904 gt 1) is

homogeneous of degree zero 1 lt 1199041015840lt 119901 lt 119899120573 1119902 =

1119901 minus120573119899 0 lt 120581 lt 1199011199021199081199041015840

isin 119860(1199011199041015840 119902119904

1015840)119863120574119860 isin Λ

120573(|120574| =

119898 minus 1) then

10038171003817100381710038171003817119879119860

Ω11989110038171003817100381710038171003817119871119902120581119902119901(119908119902)

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) (20)

10038171003817100381710038171003817119872119860

Ω11989110038171003817100381710038171003817119871119902120581119902119901(119908119902)

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) (21)

Theorem 10 If 0 lt 120572 lt 119899 Ω isin 119871119904(119878119899minus1) (119904 gt 1) is homo-

geneous of degree zero 1 lt 1199041015840 lt 119901 lt 119899120572 1119902 = 1119901 minus 1205721198990 lt 120581 lt 119901119902 119908119904

1015840

isin 119860(1199011199041015840 119902119904

1015840) 119863120574119860 isin BMO (|120574| = 119898 minus 1)

then

10038171003817100381710038171003817119879119860

Ω12057211989110038171003817100381710038171003817119871119902120581119902119901(119908119902)

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) (22)

10038171003817100381710038171003817119872119860

Ω12057211989110038171003817100381710038171003817119871119902120581119902119901(119908119902)

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) (23)

When 119898 = 1 and 119898 = 2 we denote 119879119860Ω119872119860

Ωby [119860 119879

Ω]

[119860119872Ω] and

119860

Ω 119860

Ω respectively in order to distinguish

from119879119860Ωand119872119860

Ωthat are defined for any119898 isin Nlowast To bemore

precise

[119860 119879Ω] 119891 (119909) = int

R119899

Ω(119909 minus 119910)

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899(119860 (119909) minus 119860 (119910))

sdot 119891 (119910) 119889119910

[119860119872Ω] 119891 (119909) = sup

119903gt0

1

119903119899int|119909minus119910|lt119903

Ω(119909 minus 119910)

sdot (119860 (119909) minus 119860 (119910)) 119891 (119910) 119889119910

119860

Ω119891 (119909) = int

R119899

Ω(119909 minus 119910)

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899+1(119860 (119909) minus 119860 (119910) minus nabla119860 (119910)

sdot (119909 minus 119910)) 119891 (119910) 119889119910

119860

Ω119891 (119909) = sup

119903gt0

1

119903119899+1int|119909minus119910|lt119903

Ω(119909 minus 119910)

sdot (119860 (119909) minus 119860 (119910) minus nabla119860 (119910) (119909 minus 119910)) 119891 (119910) 119889119910

(24)

Then for the above operators we have the following resultson weighted Morrey spaces with one weight

Theorem 11 IfΩ isin 119871119904(119878119899minus1) (119904 gt 1) is homogeneous of degreezero and satisfies the vanishing condition int

119878119899minus1Ω(119909

1015840)119889120590(119909

1015840) =

0 1 lt 1199041015840 lt 119901 lt infin 0 lt 120581 lt 1 119908 isin 1198601199011199041015840 119860 isin BMO then

1003817100381710038171003817[119860 119879Ω] 1198911003817100381710038171003817119871119901120581(119908) le 119862 119860lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908) (25)

1003817100381710038171003817[119860119872Ω] 1198911003817100381710038171003817119871119901120581(119908) le 119862 119860lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908) (26)


Theorem 12 If Ω isin 119871infin(119878119899minus1) is homogeneous of degree zeroand satisfies the moment condition int

119878119899minus1120579Ω(120579)119889120579 = 0 1 lt 119901 lt

infin 0 lt 120581 lt 1 119908 isin 119860119901 nabla119860 isin BMO then

100381710038171003817100381710038171003817119860

Ω119891100381710038171003817100381710038171003817119871119901120581(119908)

le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908) (27)

100381710038171003817100381710038171003817119860

Ω119891100381710038171003817100381710038171003817119871119901120581(119908)

le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908) (28)

Remark 13 Here we point out that for 119879119860Ωand 119872119860

Ω when

119863120574119860 isin BMO (|120574| = 119898 minus 1) the analogues of Theorems 11

and 12 are open for119898 ge 3

Remark 14 Define

1198791198601119860119896

Ω120572119891 (119909) = int

R119899

119896

prod

119894=1

119877119898119894

(119860119894 119909 119910)

sdotΩ (119909 minus 119910)

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572+119873119891 (119910) 119889119910

1198721198601119860119896

Ω120572119891 (119909) = sup

119903gt0

1

119903119899minus120572+119873int|119909minus119910|lt119903

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

sdot

119896

prod

119894=1

10038161003816100381610038161003816119877119898119894

(119860119894 119909 119910)

10038161003816100381610038161003816

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

(29)

where 119877119898119894

(119860119894 119909 119910) = 119860

119894(119909) minus sum

|120574|lt119898119894

(1120574)119863120574119860119894(119910)(119909 minus

119910)120574 119894 = 1 119896 119873 = sum

119896

119894=1(119898119894minus 1) When 0 lt 120572 lt 119899

they are a class of multilinear fractional integral operatorsand multilinear fractional maximal operators When 120572 = 0they are a class of multilinear singular integral operators andmultilinear maximal singular integral operators Repeatingthe proofs of the theorems above we will find that for1198791198601119860119896

Ω120572and 119872119860

1119860119896

Ω120572 the conclusions of Theorems 8 and

9 above with the bounds 119862prod119896119894=1(sum|120574|=119898

119894minus1119863120574119860119894Λ120573

) andTheorem 10 with the bounds 119862prod119896

119894=1(sum|120574|=119898

119894minus1119863120574119860119894lowast) also

hold respectively

The organization of this paper is as follows In Section 2we give some requisite lemmas and well-known results thatare important in proving the theorems The proof of thetheorems will be shown in Section 3

2 Lemmas and Well-Known Results

Lemma 15 (see [1]) Let 119860(119909) be a function on R119899 with 119898thorder derivatives in 119871119897loc(R

119899) for some 119897 gt 119899 Then

1003816100381610038161003816119877119898 (119860 119909 119910)1003816100381610038161003816

le 1198621003816100381610038161003816119909 minus 119910

1003816100381610038161003816

119898sum

|120574|=119898

(1

1003816100381610038161003816119868119910

119909

1003816100381610038161003816

int119868119910

119909

1003816100381610038161003816119863120574119860 (119911)

1003816100381610038161003816

119897119889119911)

1119897

(30)

where 119868119910119909is the cube centered at 119909with sides parallel to the axes

whose diameter is 5radic119899|119909 minus 119910|

Lemma 16 (see [12]) For 0 lt 120573 lt 1 1 le 119902 lt infin we have

10038171003817100381710038171198911003817100381710038171003817Λ120573

asymp sup119876

1

|119876|1+120573119899

int119876

1003816100381610038161003816119891 (119909) minus 119887119876 (119891)1003816100381610038161003816 119889119909

asymp sup119876

1

|119876|120573119899(1

|119876|int119876

1003816100381610038161003816119891 (119909) minus 119887119876 (119891)1003816100381610038161003816

119902119889119909)

1119902

(31)

For 119902 = infin the formula should be interpreted appropriately

Lemma 17 (see [13]) Let 1198761sub 119876

2 119892 isin Λ

120573(0 lt 120573 lt 1) Then

100381610038161003816100381610038161198921198761

minus 1198921198762

10038161003816100381610038161003816le 119862

100381610038161003816100381611987621003816100381610038161003816

120573119899 10038171003817100381710038171198921003817100381710038171003817Λ120573

(32)

Theorem 18 (see [14]) Suppose that 0 lt 120572 lt 119899 1 lt 119901 lt 1198991205721119902 = 1119901 minus 120572119899 and Ω isin 119871119904(119878119899minus1) (119904 gt 1) is homogeneous ofdegree zero Then 119879

Ω120572is a bounded operator from 119871

119901(119908119901) to

119871119902(119908119902) if the index set 120572 119901 119902 119904 satisfies one of the following

conditions

(a) 1199041015840 lt 119901 and 119908(119909)1199041015840

isin 119860(1199011199041015840 119902119904

1015840)

(b) 119904 gt 119902 and 119908(119909)minus1199041015840

isin 119860(11990210158401199041015840 11990110158401199041015840)

(c) 120572119899+1119904 lt 1119901 lt 11199041015840 and there is 119903 1 lt 119903 lt 119904(119899120572)1015840

such that 119908(119909)1199031015840

isin 119860(119901 119902)

Lemma 19 (see [10]) If 119908 isin Δ2 then there exists a constant

1198631gt 1 such that

119908 (2119861) ge 1198631119908 (119861) (33)

We call1198631the reverse doubling constant

Theorem 20 (see [4]) Suppose that 0 lt 120572 lt 119899 1 lt 119901 lt 1198991205721119902 = 1119901 minus 120572119899 Ω isin 119871

119904(119878119899minus1) (119904 gt 1) is homogeneous of

degree zero Moreover for 1 le 119894 le 119896 |120574| = 119898119894minus 1119898

119894ge 2 and

119863120574119860119894isin BMO(R119899) if the index set 120572 119901 119902 119904 satisfies one of

the following conditions

(a) 1199041015840 lt 119901 and 119908(119909)1199041015840

isin 119860(1199011199041015840 119902119904

1015840)

(b) 119904 gt 119902 and 119908(119909)minus1199041015840

isin 119860(11990210158401199041015840 11990110158401199041015840)

(c) 120572119899+1119904 lt 1119901 lt 11199041015840 and there is 119903 1 lt 119903 lt 119904(119899120572)1015840such that 119908(119909)119903

1015840

isin 119860(119901 119902)

Then there is a 119862 gt 0 independent of 119891 and 119860119894 such that

(intR119899

100381610038161003816100381610038161198791198601119860119896

Ω120572119891 (119909)119908 (119909)

10038161003816100381610038161003816

119902

119889119909)

1119902

le 119862

119896

prod

119894=1

( sum

|120574|=119898119894minus1

1003817100381710038171003817119863120574119860119894

1003817100381710038171003817lowast)

sdot (intR119899

1003816100381610038161003816119891 (119909)119908 (119909)1003816100381610038161003816

119901119889119909)

1119901

(34)

Lemma 21 (see [15]) (a) (John-Nirenberg Lemma) Let 1 le119901 lt infin Then 119887 isin BMO if and only if

1

|119876|int119876

1003816100381610038161003816119887 minus 1198871198761003816100381610038161003816

119901119889119909 le 119862 119887

119901

lowast (35)


(b) Assume 119887 isin BMO then for cubes 1198761sub 119876

2

100381610038161003816100381610038161198871198761

minus 1198871198762

10038161003816100381610038161003816le 119862 log(

100381610038161003816100381611987621003816100381610038161003816

100381610038161003816100381611987611003816100381610038161003816

) 119887lowast (36)

(c) If 119887 isin BMO then10038161003816100381610038161198872119895+1119861 minus 119887119861

1003816100381610038161003816 le 2119899(119895 + 1) 119887lowast (37)

Theorem 22 (see [16]) Suppose that Ω isin 119871119904(119878119899minus1) (119904 gt

1) is homogeneous of degree zero and satisfies the vanishingcondition int

119878119899minus1Ω(119909

1015840)119889120590(119909

1015840) = 0 If 119887 isin BMO(R119899) then [119887 119879

Ω]

is bounded on 119871119901(119908) if the index set 119901 119902 119904 satisfies one of thefollowing conditions

(a) 1199041015840 le 119901 lt infin 119901 = 1 and 119908 isin 1198601199011199041015840

(b) 1 le 119901 le 119904 119901 = infin and 1199081minus1199011015840

isin 11986011990110158401199041015840

(c) 1 le 119901 lt infin and 1199081199041015840

isin 119860119901

Theorem 23 (see [2]) If Ω isin 119871infin(119878119899minus1) is homogeneous of

degree zero and satisfies themoment condition int119878119899minus1120579Ω(120579)119889120579 =

0 119908 isin 119860119901 1 lt 119901 lt infin nabla119860 isin BMO then we have100381710038171003817100381710038171003817119860

Ω119891100381710038171003817100381710038171003817119871119901(119908)

le 119862 Ωinfin nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901(119908) (38)

Lemma 24 (see [15]) The following are true(1) If 119908 isin 119860

119901for some 1 le 119901 lt infin then 119908 isin Δ

2 More

precisely for all 120582 gt 1 we have

119908 (120582119876) le 119862120582119899119901119908 (119876) (39)

(2) If 119908 isin 119860119901for some 1 le 119901 lt infin then there exist 119862 gt 0

and 120575 gt 0 such that for any cube 119876 and a measurableset 119878 sub 119876

119908 (119878)

119908 (119876)le 119862(

|119878|

|119876|)

120575

(40)

Lemma25 (see [17]) Let119908 isin 119860infinThen the normofBMO(119908)

is equivalent to the norm of BMO(R119899) where

BMO (119908) = 119887 119887lowast119908

= sup119876

1

119908 (119876)int119876

1003816100381610038161003816119887 (119909) minus 1198981198761199081198871003816100381610038161003816 119908 (119909) 119889119909

119898119876119908119887 =

1

119908 (119876)int119876

119887 (119909)119908 (119909) 119889119909

(41)

3 Proofs of the Main Results

Before proving Theorem 8 we give a pointwise estimate of119879119860

Ω120572119891(119909) at first Set

119879Ω120572+120573

119891 (119909) = intR119899

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572minus120573

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

0 lt 120572 + 120573 lt 119899

(42)

whereΩ isin 119871119904(119878119899minus1) (119904 gt 1) is homogeneous of degree zero inR119899 Then we have the following estimate

Theorem26 If 120572 ge 0 0 lt 120572+120573 lt 119899119863120574119860 isin Λ120573(|120574| = 119898minus1)

then there exists a constant 119862 independent of 119891 such that

10038161003816100381610038161003816119879119860

Ω120572119891 (119909)

10038161003816100381610038161003816le 119862( sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573

)119879Ω120572+120573

119891 (119909) (43)

Proof For fixed 119909 isin R119899 119903 gt 0 let 119876 be a cube with center at119909 and diameter 119903 Denote 119876

119896= 2

119896119876 and set

119860119876119896

(119910) = 119860 (119910) minus sum

|120574|=119898minus1

1

120574119898119876119896

(119863120574119860)119910

120574 (44)

where 119898119876119896

119891 is the average of 119891 on 119876119896 Then we have when

|120574| = 119898 minus 1

119863120574119860119876119896

(119910) = 119863120574119860 (119910) minus 119898

119876119896

(119863120574119860) (45)

and it is proved in [1] that

119877119898(119860 119909 119910) = 119877

119898(119860119876119896

119909 119910) (46)

Hence

10038161003816100381610038161003816119879119860

Ω120572119891 (119909)

10038161003816100381610038161003816

le

infin

sum

119896=minusinfin

int2119896minus1119903le|119909minus119910|lt2

119896119903

10038161003816100381610038161003816119877119898(119860119876119896

119909 119910)10038161003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119898minus1

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

flinfin

sum

119896=minusinfin

119879119896

(47)

By Lemma 15 we get

10038161003816100381610038161003816119877119898(119860119876119896

119909 119910)10038161003816100381610038161003816

le10038161003816100381610038161003816119877119898minus1

(119860119876119896

119909 119910)10038161003816100381610038161003816

+ 119862 sum

|120574|=119898minus1

10038161003816100381610038161003816119863120574119860119876119896

(119910)10038161003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119898minus1

le 1198621003816100381610038161003816119909 minus 119910

1003816100381610038161003816

119898minus1sum

|120574|=119898minus1

(1

1003816100381610038161003816119868119910

119909

1003816100381610038161003816

int119868119910

119909

10038161003816100381610038161003816119863120574119860119876119896

(119911)10038161003816100381610038161003816

119897

119889119911)

1119897

+ 1198621003816100381610038161003816119909 minus 119910

1003816100381610038161003816

119898minus1sum

|120574|=119898minus1

10038161003816100381610038161003816119863120574119860119876119896

(119910)10038161003816100381610038161003816

(48)


Note that if |119909 minus 119910| lt 2119896119903 then 119868119910119909sub 5119899119876

119896 By Lemmas 16

and 17 we have when |120574| = 119898 minus 1

(1

1003816100381610038161003816119868119910

119909

1003816100381610038161003816

int119868119910

119909

10038161003816100381610038161003816119863120574119860119876119896

(119911)10038161003816100381610038161003816

119897

119889119911)

1119897

= (1

1003816100381610038161003816119868119910

119909

1003816100381610038161003816

int119868119910

119909

10038161003816100381610038161003816119863120574119860 (119911) minus 119898119876

119896

(119863120574119860)10038161003816100381610038161003816

119897

119889119911)

1119897

le (1

1003816100381610038161003816119868119910

119909

1003816100381610038161003816

int119868119910

119909

10038161003816100381610038161003816119863120574119860 (119911) minus 119898

119868119910

119909

(119863120574119860)10038161003816100381610038161003816

119897

119889119911)

1119897

+10038161003816100381610038161003816119898119868119910

119909

(119863120574119860) minus 119898

5119899119876119896

(119863120574119860)10038161003816100381610038161003816

+100381610038161003816100381610038161198985119899119876119896

(119863120574119860) minus 119898

119876119896

(119863120574119860)10038161003816100381610038161003816

le 11986210038161003816100381610038161198761198961003816100381610038161003816

120573119899 10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573

le 119862 (2119896119903)120573 1003817100381710038171003817119863

1205741198601003817100381710038171003817Λ120573

(49)

It is obvious that when |120574| = 119898 minus 1

10038161003816100381610038161003816119863120574119860119876119896

(119910)10038161003816100381610038161003816=10038161003816100381610038161003816119863120574119860 (119910) minus 119898

119876119896

(119863120574119860)10038161003816100381610038161003816

le 11986210038161003816100381610038161198761198961003816100381610038161003816

120573119899 10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573

le 119862 (2119896119903)120573 1003817100381710038171003817119863

1205741198601003817100381710038171003817Λ120573

(50)

Thus

10038161003816100381610038161003816119877119898(119860119876119896

119909 119910)10038161003816100381610038161003816

le 1198621003816100381610038161003816119909 minus 119910

1003816100381610038161003816

119898minus1(2119896119903)120573

sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573

(51)

Therefore

119879119896le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573

sdot int2119896minus1119903le|119909minus119910|lt2

119896119903

(2119896119903)120573

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572

1003816100381610038161003816Ω (119909 minus 119910)10038161003816100381610038161003816100381610038161003816119891 (119910)

1003816100381610038161003816 119889119910

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573


119896119903

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572minus120573

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

(52)

It follows that

10038161003816100381610038161003816119879119860

Ω120572119891 (119909)

10038161003816100381610038161003816le

infin

sum

119896=minusinfin

(119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573


119896119903

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572minus120573

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910)

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573

sdot

infin

sum

119896=minusinfin


119896119903

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572minus120573

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

= 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573

intR119899

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572minus120573

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

= 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573

119879Ω120572+120573

119891 (119909)

(53)

Thus we finish the proof of Theorem 26

The following theorem is a key theorem in proving (18) ofTheorem 8

Theorem 27 Under the same assumptions of Theorem 8119879Ω120572+120573

is bounded from 119871119901120581(119908119901 119908119902) to 119871119902120581119902119901(119908119902)

Proof Fix a ball 119861(1199090 119903119861) we decompose 119891 = 119891

1+ 119891

2with

1198911= 119891120594

2119861 Then we have

(1

119908119902 (119861)120581119902119901

int119861

10038161003816100381610038161003816119879Ω120572+120573

119891 (119909)10038161003816100381610038161003816

119902

119908119902(119909) 119889119909)

1119902

le1

119908119902 (119861)120581119901(int119861

10038161003816100381610038161003816119879Ω120572+120573

1198911 (119909)

10038161003816100381610038161003816

119902

119908119902(119909) 119889119909)

1119902

+1

119908119902 (119861)120581119901(int119861

10038161003816100381610038161003816119879Ω120572+120573

1198912 (119909)

10038161003816100381610038161003816

119902

119908119902(119909) 119889119909)

1119902

fl 1198691+ 1198692

(54)

We estimate 1198691at first By Remark 7(a) we know that119908119902 isin Δ

2

Then byTheorem 18(a) and the fact that 119908119902 isin Δ2we get

1198691le

1

119908119902 (119861)120581119901

10038171003817100381710038171003817119879Ω120572+120573

1198911

10038171003817100381710038171003817119871119902(119908119902)

le119862

119908119902 (119861)120581119901

100381710038171003817100381711989111003817100381710038171003817119871119901(119908119901)

=119862

119908119902 (119861)120581119901(int2119861

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119901119908 (119909)

119901119889119909)

1119901

le 11986210038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

119908119902(2119861)

120581119901

119908119902 (119861)120581119901

le 11986210038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

(55)


Now we consider the term 1198692 By Holderrsquos inequality we have

10038161003816100381610038161003816119879Ω120572+120573

1198912 (119909)

10038161003816100381610038161003816=

infin

sum

119895=1

int2119895+11198612119895119861

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572minus120573

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

le 119862

infin

sum

119895=1

(int2119895+1119861

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

119904119889119910)

1119904

sdot (int2119895+11198612119895119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816

1199041015840

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

(119899minus120572minus120573)1199041015840119889119910)

11199041015840

fl 119862infin

sum

119895=1

(11986811198951198682119895)

(56)

We will estimate 1198681119895 1198682119895 respectively Let 119911 = 119909 minus 119910 then for

119909 isin 119861 119910 isin 2119895+1119861 we have 119911 isin 2119895+2119861 Noticing that Ω ishomogeneous of degree zero andΩ isin 119871119904(119878119899minus1) then we have

1198681119895= (int

2119895+2119861

|Ω (119911)|119904119889119911)

1119904

= (int

2119895+2

119903119861

0

int119878119899minus1

10038161003816100381610038161003816Ω (119911

1015840)10038161003816100381610038161003816

119904

1198891199111015840119903119899minus1119889119903)

1119904

= 119862 Ω119871119904(119878119899minus1)

100381610038161003816100381610038162119895+211986110038161003816100381610038161003816

1119904

(57)

where 1199111015840 = 119911|119911| For 119909 isin 119861 119910 isin (2119861)119888 we have |119909 minus 119910| sim|1199090minus 119910| Thus

1198682119895le

119862

10038161003816100381610038162119895+1119861

1003816100381610038161003816

1minus(120572+120573)119899(int2119895+1119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816

1199041015840

119889119910)

11199041015840

(58)

By Holderrsquos inequality and 1199081199041015840

isin 119860(1199011199041015840 119902119904

1015840) we get

(int2119895+1119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816

1199041015840

119889119910)

11199041015840

le 119862(int2119895+1119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119901119908 (119910)

119901119889119910)

1119901

sdot (int2119895+1119861

119908 (119910)minus1199011199041015840

(119901minus1199041015840

)119889119910)

(119901minus1199041015840

)1199011199041015840

le 11986210038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) 119908

119902(2119895+1119861)120581119901

sdot (int2119895+1119861

119908 (119910)minus1199011199041015840

(119901minus1199041015840

)119889119910)

(119901minus1199041015840

)1199011199041015840

le 11986210038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) 119908

119902(2119895+1119861)120581119901

sdot

100381610038161003816100381610038162119895+111986110038161003816100381610038161003816

(119901119902minus1199041015840

119902+1199041015840

119901)1199011199021199041015840

119908119902 (2119895+1119861)1119902

(59)

Thus

10038161003816100381610038161003816119879Ω120572+120573

1198912 (119909)

10038161003816100381610038161003816le 119862

infin

sum

119895=1

(11986811198951198682119895)

le 119862

infin

sum

119895=1

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

1

119908119902 (2119895+1119861)1119902minus120581119901

(60)

So we get

1198692le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

infin

sum

119895=1

119908119902(119861)

1119902minus120581119901

119908119902 (2119895+1119861)1119902minus120581119901

(61)

We know from Remark 7(a) and Lemma 19 that 119908119902 satisfiesinequality (33) so the above series converges since the reversedoubling constant is larger than one Hence

1198692le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) (62)

Therefore the proof of Theorem 27 is completed

Remark 28 It is worth noting that Theorem 27 is essentiallyverifying the multilinear fractional operator 119879

Ω120572is bounded

on weighted Morrey spaces

Now we are in the position of provingTheorem 8We will obtain (18) immediately in combination ofTheo-

rems 26 and 27Then let us turn to prove (19)Set

119879119860

Ω120572119891 (119909)

= intR119899

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572+119898minus1

1003816100381610038161003816119877119898 (119860 119909 119910)10038161003816100381610038161003816100381610038161003816119891 (119910)

1003816100381610038161003816 119889119910

0 le 120572 lt 119899

(63)

whereΩ isin 119871119904(119878119899minus1) (119904 gt 1) is homogeneous of degree zero inR119899 It is easy to see inequality (18) also holds for 119879

119860

Ω120572 On the

other hand for any 119903 gt 0 we have

119879119860

Ω120572119891 (119909)

ge int|119909minus119910|lt119903

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572+119898minus1

1003816100381610038161003816119877119898 (119860 119909 119910)10038161003816100381610038161003816100381610038161003816119891 (119910)

1003816100381610038161003816 119889119910

ge1

119903119899minus120572+119898minus1


1003816100381610038161003816Ω (119909 minus 119910)10038161003816100381610038161003816100381610038161003816119877119898 (119860 119909 119910)

10038161003816100381610038161003816100381610038161003816119891 (119910)

1003816100381610038161003816 119889119910

(64)

Taking the supremum for 119903 gt 0 on the inequality above weget

119879119860

Ω120572119891 (119909) ge 119872

119860

Ω120572119891 (119909) (65)

Thus we can immediately obtain (19) from (65) and (18)


Similarly as before we give the following theorem at firstbefore provingTheorem 9 since it plays an important role inthe proof of Theorem 9 Set

119879Ω120573119891 (119909) = int

R119899

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120573

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910 (66)

whereΩ isin 119871119904(119878119899minus1) (119904 gt 1) is homogeneous of degree zero inR119899

Theorem 29 Under the assumptions of Theorem 9 119879Ω120573

isbounded from 119871119901120581(119908119901 119908119902) to 119871119902120581119902119901(119908119902)

The proof of Theorem 29 can be treated as that ofTheorem 27 with only slight modifications we omit its proofhere

Now let us prove Theorem 9 It is not difficult to see that(20) can be easily obtained from Theorems 26 and 29 Thenwe can immediately arrive at (21) from (65) and (20)

From now on we are in the place of showingTheorem 10We prove (22) at first Fixing any cube119876 with center at 119909 anddiameter 119903 denote 119876 = 2119876 and set

119860119876(119910) = 119860 (119910) minus sum

|120574|=119898minus1

1

120574119898119876(119863120574119860)119910

120574 (67)

Noticing that equality (67) is the special case of equality (44)when 119896 = 1 Thus equalities (45) and (46) also hold for119860119876(119910) We decompose 119891 as 119891 = 119891120594

119876+ 119891120594

(119876)119888 = 119891

1+ 119891

2

Then we have

1

119908119902 (119876)120581119901(int119876

10038161003816100381610038161003816119879119860

Ω120572119891 (119910)

10038161003816100381610038161003816

119902

119908 (119910)119902119889119910)

1119902

le1

119908119902 (119876)120581119901(int119876

10038161003816100381610038161003816119879119860

Ω1205721198911(119910)10038161003816100381610038161003816

119902

119908 (119910)119902119889119910)

1119902

+1

119908119902 (119876)120581119901(int119876

10038161003816100381610038161003816119879119860

Ω1205721198912(119910)10038161003816100381610038161003816

119902

119908 (119910)119902119889119910)

1119902

fl 119868 + 119868119868

(68)

ByTheorem 20(a) and Remark 7(a) that 119908119902 isin Δ2 we have

119868 le119862

119908119902 (119876)120581119901

sdot sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast (int

119876

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119901119908 (119910)

119901119889119910)

1119901

= 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)(

119908119902(119876)

119908119902 (119876))

120581119901

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

(69)

Next we consider the term 119879119860

Ω1205721198912(119910) contained in 119868119868 By

Lemma 15 and equality (45) (46) we have

10038161003816100381610038161003816119879119860

Ω1205721198912(119910)10038161003816100381610038161003816

le int(119876)119888

10038161003816100381610038161003816119877119898(119860119876 119910 119911)

100381610038161003816100381610038161003816100381610038161003816119910 minus 119911

1003816100381610038161003816

119898minus1

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

le 119862int(119876)119888

sum

|120574|=119898minus1

(1

10038161003816100381610038161003816119868119911119910

10038161003816100381610038161003816

int119868119911

119910

10038161003816100381610038161003816119863120574119860119876(119905)10038161003816100381610038161003816

119897

119889119905)

1119897

sdot1003816100381610038161003816Ω (119910 minus 119911)

1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572119889119911

+ 119862int(119876)119888

sum

|120574|=119898minus1

10038161003816100381610038161003816119863120574119860 (119911) minus 119898

119876(119863120574119860)10038161003816100381610038161003816

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

sdot

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572119889119911 fl 119868119868

1+ 119868119868

2

(70)

We estimate 1198681198681and 119868119868

2 respectively By Lemma 21(a) and (b)

Holderrsquos inequality and 1199081199041015840

isin 119860(1199011199041015840 119902119904

1015840) we get

1198681198681le 119862 sum

|120574|=119898minus1

(1

10038161003816100381610038161003816119868119911119910

10038161003816100381610038161003816

int119868119911

119910

10038161003816100381610038161003816119863120574119860 (119905) minus 119898

119876(119863120574119860)10038161003816100381610038161003816

119897

119889119905)

1119897

sdot

infin

sum

119895=1

int2119895+11198762119895119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572119889119911

le 119862 sum

|120574|=119898minus1

[

[

(1

10038161003816100381610038161003816119868119911119910

10038161003816100381610038161003816

sdot int119868119911

119910

100381610038161003816100381610038161003816119863120574119860 (119905) minus 119898119868119911

119910

(119863120574119860)100381610038161003816100381610038161003816

119897

119889119905)

1119897

+100381610038161003816100381610038161003816119898119868119911

119910

(119863120574119860)

minus 1198985119899119876(119863120574119860)100381610038161003816100381610038161003816+100381610038161003816100381610038161198985119899119876(119863120574119860) minus 119898

119876(119863120574119860)10038161003816100381610038161003816]

]

sdot

infin

sum

119895=1

(int2119895+1119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

119904119889119911)

1119904

sdot (int2119895+11198762119895119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

(119899minus120572)1199041015840119889119911)

11199041015840

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast Ω119871

119904(119878119899minus1)

sdot

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus120572119899(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840


le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus120572119899(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

119901

sdot 119908 (119911)119901119889119911)

1119901

sdot (int2119895+1119876

119908 (119911)minus1199011199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

sdot

infin

sum

119895=1

1

119908119902 (2119895+1119876)1119902minus120581119901

(71)

For 119910 isin 119876 119911 isin (119876)119888 we have |119910 minus 119911| sim |119909 minus 119911| so we obtain

1198681198682le 119862

infin

sum

119895=1

int2119895+11198762119895119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

sdot1003816100381610038161003816Ω (119910 minus 119911)

1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572119889119911

+ 119862

infin

sum

119895=1

int2119895+11198762119895119876

sum

|120574|=119898minus1

100381610038161003816100381610038161198982119895+1119876(119863120574119860) minus 119898

119876(119863120574119860)10038161003816100381610038161003816

sdot1003816100381610038161003816Ω (119910 minus 119911)

1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572119889119911 fl 119868119868

21+ 119868119868

22

(72)

By Holderrsquos inequality we get

11986811986821le 119862

infin

sum

119895=1

1

100381610038161003816100381621198951198761003816100381610038161003816

1minus

120572

119899

int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863

120574119860)10038161003816100381610038161003816100381610038161003816Ω (119910 minus 119911)

10038161003816100381610038161003816100381610038161003816119891 (119911)

1003816100381610038161003816 119889119911

le 119862

infin

sum

119895=1

1

100381610038161003816100381621198951198761003816100381610038161003816

1minus

120572

119899

(int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

1199041015840

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840

(int2119895+1119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

119904119889119911)

1119904

le 119862 Ω119871119904(119878119899minus1)

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus

120572

119899

(int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

1199041015840

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840

le 119862 Ω119871119904(119878119899minus1)

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus

120572

119899

(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

119901119908 (119911)

119901119889119911)

1119901

sdot (int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

1199011199041015840

(119901minus1199041015840

)119908 (119911)

minus1199011199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

(73)

We estimate the part containing the function119863120574119860 as follows

(int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

1199011199041015840

(119901minus1199041015840

)119908 (119911)

minus1199011199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

le 119862(int2119895+1119876

sum

|120574|=119898minus1

10038161003816100381610038161003816119863120574119860 (119911) minus 119898

2119895+1119876119908minus1199011199041015840(119901minus1199041015840)(119863120574119860)10038161003816100381610038161003816

1199011199041015840

(119901minus1199041015840

)

119908 (119911)minus1199011199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

+ sum

|120574|=119898minus1

100381610038161003816100381610038161198982119895+1119876119908minus1199011199041015840(119901minus1199041015840)(119863120574119860) minus 119898

2119895+1119876(119863120574119860)10038161003816100381610038161003816119908minus1199011199041015840

(119901minus1199041015840

)(2119895+1119876)(119901minus1199041015840

)1199011199041015840

fl 119868119868119868 + 119868119881

(74)

For the term 119868119868119868 since 1199081199041015840

isin 119860(1199011199041015840 119902119904

1015840) we then have

119908minus1199011199041015840

(119901minus1199041015840

)isin 119860

1199051015840 sub 119860

infinby Remark 7(b) Thus by Lemma 25

that the norm of BMO(119908minus1199011199041015840

(119901minus1199041015840

)) is equivalent to the norm

of BMO(R119899) and 1199081199041015840

isin 119860(1199011199041015840 119902119904

1015840) we have


119868119868119868 le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast 119908

minus1199011199041015840

(119901minus1199041015840

)(2119895+1119876)(119901minus1199041015840

)1199011199041015840

= 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

100381610038161003816100381610038162119895+111987610038161003816100381610038161003816

(119901minus1199041015840

)1199011199041015840

+1119902

119908119902 (2119895+1119876)1119902

(75)

For the term 119868119881 by Lemma 21(a) there exist 1198621 1198622gt 0 such

that for any cube 119876 and 119904 gt 0100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119905 isin 2119895+1119876 sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816 gt 119904

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le 1198621

100381610038161003816100381610038162119895+111987610038161003816100381610038161003816119890minus1198622119904(sum|120574|=119898minus1

119863120574

119860lowast)

(76)

sincesum|120574|=119898minus1

(119863120574119860) isin BMOThen by Lemma 24(2) we have

119908(

119905 isin 2119895+1119876 sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

gt 119904

) le 119862119908(2119895+1119876) 119890

minus1198622119904120575(sum

|120574|=119898minus1119863120574

119860lowast)

(77)

for some 120575 gt 0 Hence it implies

sum

|120574|=119898minus1


2119895+1119876(119863120574119860)10038161003816100381610038161003816

le1

119908minus1199011199041015840(119901minus119904

1015840) (2119895+1119876)

int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119905)

minus 1198982119895+1119876(119863120574119860)1003816100381610038161003816 119908minus1199011199041015840

(119901minus1199041015840

)(119905) 119889119905

=119862

119908minus1199011199041015840(119901minus119904

1015840) (2119895+1119876)

int

infin

0

119908minus1199011199041015840

(119901minus1199041015840

)(

119905

isin 2119895+1119876 sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

gt 119904

)119889119904 le119862

119908minus1199011199041015840(119901minus119904

1015840) (2119895+1119876)

sdot int

infin

0

119908minus1199011199041015840

(119901minus1199041015840

)(2119895+1119876) 119890

minus1198622119904120575(sum

|120574|=119898minus1119863120574

119860lowast)119889119904

= 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

(78)

As a result

119868119881 le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast 119908

minus1199011199041015840

(119901minus1199041015840

)(2119895+1119876)(119901minus1199041015840

)1199011199041015840

= 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

100381610038161003816100381610038162119895+111987610038161003816100381610038161003816

(119901minus1199041015840

)1199011199041015840

+1119902

119908119902 (2119895+1119876)1119902

(79)

Thus

11986811986821le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

sdot

infin

sum

119895=1

1

119908119902 (2119895+1119876)1119902minus120581119901

(80)

For the term 11986811986822 by Lemma 21(c) Holderrsquos inequality and

1199081199041015840

isin 119860(1199011199041015840 119902119904

1015840) we get

11986811986822le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

sdot

infin

sum

119895=1

119895 int2119895+11198762119895119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572119889119911

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast Ω119871

119904(119878119899minus1)

infin

sum

119895=1

119895

sdot

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus120572119899(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

infin

sum

119895=1

119895

sdot

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus120572119899(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

119901119908 (119911)

119901119889119911)

1119901

sdot (int2119895+1119876

119908 (119911)minus1199011199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

sdot

infin

sum

119895=1

119895

119908119902 (2119895+1119876)1119902minus120581119901

(81)

Hence

10038161003816100381610038161003816119879119860

Ω1205721198912(119910)10038161003816100381610038161003816le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

sdot

infin

sum

119895=1

119895

119908119902 (2119895+1119876)1119902minus120581119901

(82)

Therefore

119868119868 le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

sdot

infin

sum

119895=1

119895119908119902(119876)

1119902minus120581119901

119908119902 (2119895+1119876)1119902minus120581119901


le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

infin

sum

119895=1

119895

(119863119895+1

1)1119902minus120581119901

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

(83)

where1198631gt 1 is the reverse doubling constant Consequently

1

119908119902 (119876)120581119901(int119876

10038161003816100381610038161003816119879119860

Ω120572119891 (119910)

10038161003816100381610038161003816

119902

119908 (119910)119902119889119910)

1119902

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

(84)

Taking supremum over all cubes in R119899 on both sides ofthe above inequality we complete the proof of (22) ofTheorem 10

It is not difficult to see that inequality (23) is easy to getfrom (22) and (65)

Proof of Theorem 11 We consider (25) firstly Let 119876 be thesame as in the proof of (22) and denote 119876 = 2119876 wedecompose 119891 as 119891 = 119891120594

119876+ 119891120594

(119876)119888 fl 119891

1+ 119891

2 Then we have

1

119908 (119876)120581119901(int119876

1003816100381610038161003816[119860 119879Ω] 119891 (119910)1003816100381610038161003816

119901119908 (119910) 119889119910)

1119901

le1

119908 (119876)120581119901(int119876

1003816100381610038161003816[119860 119879Ω] 1198911 (119910)1003816100381610038161003816

119901119908 (119910) 119889119910)

1119901

+1

119908 (119876)120581119901(int119876

1003816100381610038161003816[119860 119879Ω] 1198912 (119910)1003816100381610038161003816

119901119908 (119910) 119889119910)

1119901

fl 119868 + 119868119868

(85)

ByTheorem 22(a) and Lemma 24(1) that 119908 isin Δ2 we get

119868 le1

119908 (119876)120581119901

1003817100381710038171003817[119860 119879Ω] 11989111003817100381710038171003817119871119901(119908)

le119862

119908 (119876)120581119901119860lowast

100381710038171003817100381711989111003817100381710038171003817119871119901(119908)

= 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

119908(119876)120581119901

119908 (119876)120581119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(86)

For |[119860 119879Ω]1198912(119910)| by Holderrsquos inequality we obtain

1003816100381610038161003816[119860 119879Ω] 1198912 (119910)1003816100381610038161003816 le

infin

sum

119895=1

int2119895+11198762119895119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899

1003816100381610038161003816119860 (119910) minus 119860 (119911)1003816100381610038161003816

sdot1003816100381610038161003816119891 (119911)

1003816100381610038161003816 119889119911

le 119862

infin

sum

119895=1

1

100381610038161003816100381621198951198761003816100381610038161003816

(int2119895+1119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

119904119889119911)

1119904

sdot (int2119895+1119876

1003816100381610038161003816119860 (119910) minus 119860 (119911)1003816100381610038161003816

1199041015840

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840

le 119862 Ω119871119904

sdot

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

sdot (int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840

+ 119862 Ω119871119904

sdot

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

(int2119895+1119876

100381610038161003816100381610038161198982119895+1119876119908minus1199041015840(119901minus1199041015840)(119860) minus 119860 (119911)

10038161003816100381610038161003816

1199041015840

sdot1003816100381610038161003816119891 (119911)

1003816100381610038161003816

1199041015840

119889119911)

11199041015840

fl 1198681198681(119910) + 119868119868

2

(87)

Next we estimate 1198681198681(119910) and 119868119868

2 respectively By Holderrsquos

inequality and 119908 isin 1198601199011199041015840 we have

1

119908 (119876)120581119901(int119876

1198681198681(119910)

119901119908 (119910) 119889119910)

1119901

= 119862Ω119871119904

119908 (119876)120581119901

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

(int119876

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

119901

(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

1199011199041015840

119908 (119910) 119889119910)

1119901

le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

119908 (119876)

120581

119901

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

119908 (2119895+1119876)120581119901

119908minus1199041015840

(119901minus1199041015840

)(2119895+1119876)(119901minus1199041015840

)1199011199041015840

sdot (int119876

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

119908 (119876)120581119901

infin

sum

119895=1

1

119908 (2119895+1119876)(1minus120581)119901

(int119876

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

(88)


We estimate the part containing1198982119895+1119876119908minus1199041015840(119901minus1199041015840)(119860) as follows

(int119876

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

le (int119876

1003816100381610038161003816119860 (119910) minus 119898119876119908 (119860)1003816100381610038161003816

119901119908 (119910) 119889119910)

1119901

+10038161003816100381610038161003816119898119876119908 (119860) minus 1198982119895+1119876119908minus119904

1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816119908 (119876)

1119901

fl 119868119868119868 + 119868119881

(89)

For the term 119868119868119868 notice that 119908 isin 1198601199011199041015840 sub 119860

infin we thus get by

Lemma 25 that

119868119868119868 le 119860lowast 119908 (119876)1119901 (90)

Next we estimate 119868119881 By Lemmas 21(c) and 25 we have

10038161003816100381610038161003816119898119876119908 (119860) minus 1198982119895+1119876119908minus119904

1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

le1003816100381610038161003816119898119876119908 (119860) minus 119898119876 (119860)

1003816100381610038161003816 +1003816100381610038161003816119898119876 (119860) minus 1198982119895+1119876 (119860)

1003816100381610038161003816

+100381610038161003816100381610038161198982119895+1119876 (119860) minus 1198982119895+1119876119908minus119904

1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816le

1

119908 (119876)

sdot int119876

1003816100381610038161003816119860 (119905) minus 119898119876 (119860)1003816100381610038161003816 119908 (119905) 119889119905 + 2

119899(119895 + 1) 119860lowast

+1

119908minus1199041015840(119901minus119904

1015840) (2119895+1119876)

sdot int2119895+1119876

1003816100381610038161003816119860 (119905) minus 1198982119895+1119876 (119860)1003816100381610038161003816 119908minus1199041015840

(119901minus1199041015840

)(119905) 119889119905

le 119862 (119895 + 1) 119860lowast

(91)

Hence

119868119881 le 119862 (119895 + 1) 119860lowast 119908 (119876)1119901 (92)

As a result

1

119908 (119876)120581119901(int119876

1198681198681(119910)

119901119908 (119910) 119889119910)

1119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

infin

sum

119895=1

(119895 + 1)119908 (119876)

(1minus120581)119901

119908 (2119895+1119876)(1minus120581)119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(93)

For 1198681198682 by Holderrsquos inequality and 119908 isin 119860

1199011199041015840 we get

1198681198682le 119862 Ω119871119904

sdot

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

119901119908 (119911) 119889119911)

1119901

sdot (int2119895+1119876


10038161003816100381610038161003816

1199011199041015840

(119901minus1199041015840

)

sdot 119908 (119911)minus1199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

le 119862 Ω119871119904 119860lowast

sdot10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

119908 (2119895+1119876)120581119901

sdot 119908minus1199041015840

(119901minus1199041015840

)(2119895+1119876)(119901minus1199041015840

)1199011199041015840

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

sdot

infin

sum

119895=1

1

119908 (2119895+1119876)(1minus120581)119901

(94)

Therefore

1

119908 (119876)120581119901(int119876

119868119868119901

2119908 (119910) 119889119910)

1119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

infin

sum

119895=1

119908 (119876)(1minus120581)119901

119908 (2119895+1119876)(1minus120581)119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(95)

So far we have completed the proof of (25)

Inequality (26) can be immediately obtained from (65)and (25)

Proof ofTheorem 12 As before we prove (27) at first Assume119876 to be the same as in the proof of (22) denote 119876 = 2119876 andset

119860119876(119910) = 119860 (119910) minus 119898

119876(nabla119860) 119910 (96)

We also decompose 119891 according to 119876 119891 = 119891120594119876+ 119891120594

(119876)119888 fl

1198911+ 119891

2 Then we get

1

119908 (119876)120581119901(int119876

100381610038161003816100381610038161003816119860

Ω119891 (119910)

100381610038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

le1

119908 (119876)120581119901(int119876

100381610038161003816100381610038161003816119860

Ω1198911(119910)100381610038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

+1

119908 (119876)120581119901(int119876

100381610038161003816100381610038161003816119860

Ω1198912(119910)100381610038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

fl 119868 + 119868119868

(97)


For 119868 Theorem 23 and Lemma 24(1) imply

119868 le1

119908 (119876)120581119901

100381710038171003817100381710038171003817119860

Ω1198911

100381710038171003817100381710038171003817119871119901(119908)

le119862

119908 (119876)120581119901Ωinfin nabla119860lowast

100381710038171003817100381711989111003817100381710038171003817119871119901(119908)

le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

119908(119876)120581119901

119908 (119876)120581119901

le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(98)

We will omit the proof for 119868119868 since it is similar to and eveneasier than the part of 119868119868 in the proof of (22) except that weuse the conditions 119908 isin 119860

119901 Ω isin 119871

infin(119878119899minus1) 119898 = 2 and 119891 isin

119871119901120581(119908) For inequality (28) it can be easily proved by (27)

and (65) Thus we complete the proof of Theorem 12

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The research was partially supported by the National NatureScience Foundation of China under Grant no 11171306 andno 11571306 and sponsored by the Scientific Project ofZhejiang Provincial Science Technology Department underGrant no 2011C33012 and the Scientific Research Fund ofZhejiang Provincial Education Department under Grant noZ201017584

References

[1] J Cohen and J Gosselin ldquoA BMO estimate for multilinearsingular integralsrdquo Illinois Journal of Mathematics vol 30 no3 pp 445ndash464 1986

[2] S Hofmann ldquoOn certain nonstandard Calderon-Zygmundoperatorsrdquo StudiaMathematica vol 109 no 2 pp 105ndash131 1994

[3] S Z Lu H XWu and P Zhang ldquoMultilinear singular integralswith rough kernelrdquo Acta Mathematica Sinica (English Series)vol 19 no 1 pp 51ndash62 2003

[4] Y Ding and S Z Lu ldquoWeighted boundedness for a class ofroughmultilinear operatorsrdquoActaMathematica SinicamdashEnglishSeries vol 17 no 3 pp 517ndash526 2001

[5] S Z Lu and P Zhang ldquoLipschitz estimates for generalizedcommutators of fractional integrals with rough kernelrdquoMathe-matische Nachrichten vol 252 no 1 pp 70ndash85 2003

[6] C B Morrey ldquoOn the solutions of quasi-linear elliptic partialdifferential equationsrdquo Transactions of the AmericanMathemat-ical Society vol 43 no 1 pp 126ndash166 1938

[7] F Chiarenza and M Frasca ldquoMorrey spaces and Hardy-Littlewoodmaximal functionrdquo Rendiconti di Matematica e dellesue Applicazioni vol 7 no 3-4 pp 273ndash279 1987

[8] D Fan S Lu and D Yang ldquoRegularity in Morrey spaces ofstrong solutions to nondivergence elliptic equations with VMOcoefficientsrdquo Georgian Mathematical Journal vol 5 no 5 pp425ndash440 1998

[9] J Peetre ldquoOn the theory of 119871120588120582

spacesrdquo Journal of FunctionalAnalysis vol 4 no 1 pp 71ndash87 1969

[10] Y Komori and S Shirai ldquoWeighted Morrey spaces and asingular integral operatorrdquo Mathematische Nachrichten vol282 no 2 pp 219ndash231 2009

[11] H Wang ldquoThe boundedness of some operators with roughkernel on the weighted Morrey spacesrdquo Acta MathematicaSinica vol 55 no 4 pp 589ndash600 2012 (Chinese)

[12] M Paluszynski ldquoCharacterization of the Besov spaces viathe commutator operator of Coifman Rochberg and WeissrdquoIndiana University Mathematics Journal vol 44 no 1 pp 1ndash171995

[13] RADeVore andRC Sharpley ldquoMaximal functionsmeasuringsmoothnessrdquo Memoirs of the American Mathematical Societyvol 47 no 293 1984

[14] Y Ding and S Z Lu ldquoWeighted norm inequalities for fractionalintegral operators with rough kernelrdquo Canadian Journal ofMathematics vol 50 no 1 pp 29ndash39 1998

[15] L GrafakosModern Fourier Analysis vol 250 ofGraduate Textsin Mathematics Springer New York NY USA 2nd edition2009

[16] S Z Lu Y Ding and D Y Yan Singular Integrals aand RelatedTopics World Scientific 2007

[17] BMuckenhoupt and R LWheeden ldquoWeighted boundedmeanoscillation and the Hilbert transformrdquo StudiaMathematica vol54 no 3 pp 221ndash237 1976

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


the 119871119901 boundedness of119879119860Ωwas obtained bymeans of a good-120582

inequality by Cohen and Gosselin [1] In 1994 Hofmann [2]proved that 119879119860

Ωis a bounded operator on 119871119901(119908) when nabla119860 isin

119861119872119874 and 119908 isin 119860119901 Recently Lu et al [3] proved 119879119860

Ωand119872119860

Ω

are bounded from 119871119901 to 119871119902 (1119901 minus 1119902 = 120573119899) when 119863120574119860 isin

Λ120573(|120574| = 119898 minus 1) while for multilinear fractional integral

operators Ding and Lu [4] showed the (119871119901(119908119901) 119871119902(119908119902))boundedness of 1198791198601 119860119896

Ω120572and119872119860

1119860119896

Ω120572(their definitions will

be given later) if 119863120574119860119895isin BMO (|120574| = 119898 minus 1) 119895 = 1 119896

After that Lu and Zhang [5] proved 119879119860Ω120572

is a boundedoperator from 119871119901 to 119871119902 (1119901 minus 1119902 = (120572 + 120573)119899) when119863120574119860 isinΛ120573(|120574| = 119898 minus 1)On the other hand the classical Morrey spaces were first

introduced by Morrey [6] to study the local behavior of solu-tions to second-order elliptic partial differential equationsFrom then on a lot of works concerning Morrey spacesand some related spaces have been done see [7ndash9] and thereferences therein for details In 2009 Komori and Shirai[10] first studied the weightedMorrey spaces and investigatedsome classical singular integrals in harmonic analysis onthem such as the Hardy-Littlewood maximal operator theCalderon-Zygmund operator the fractional integral opera-tor and the fractional maximal operator Recently Wang [11]discussed the boundedness of the classical singular operatorswith rough kernels on the weighted Morrey spaces

We note that many works concerning 119879119860Ω120572

119872119860

Ω120572 119879119860Ω

and119872119860

Ωhave been done on 119871119901 spaces or weighted 119871119901 spaces

when 119863120574119860 belongs to some function spaces for |120574| = 119898 minus1 However there is not any study about these operatorson weighted Morrey spaces Therefore it is natural to askwhether they are bounded on weighted Morrey spaces Theaim of this paper is to investigate the boundedness of 119879119860

Ω120572

119872119860

Ω120572 119879119860Ω and 119872119860

Ωon weighted Morrey spaces if 119863120574119860 isin

Λ120573(|120574| = 119898 minus 1) or 119863120574119860 isin BMO (|120574| = 119898 minus 1)

When 119863120574119860 isin Λ120573(|120574| = 119898 minus 1) we show 119879

119860

Ω120572and

119879119860

Ωare controlled pointwisely by the fractional singular

integral operators 119879Ω120572+120573

and 119879Ω120573

(their definition will begiven later) respectively Thus the problem of studying theboundedness of119879119860

Ω120572and119879119860

ΩonweightedMorrey spaces with

two weights could be reduced to that of 119879Ω120572+120573

and 119879Ω120573

When 119863120574119860 isin BMO (|120574| = 119898 minus 1) the boundedness of119879119860

Ω120572on weighted Morrey spaces with two weights is proved

by standard method However we could only obtain theboundedness of 119879119860

Ωon weighted Morrey spaces with one

weight for 119898 = 1 and 119898 = 2 since we need the 119871119901(119908)boundedness of 119879119860

Ωin our proof but to the best of our

knowledge there is not such bounds hold for119879119860Ωwhen119898 ge 3

For119872119860

Ω120572and119872119860

Ω we show they are controlled pointwisely by

119879119860

Ω120572and 119879119860

Ω respectively Thus it is easy to obtain the same

results for119872119860

Ω120572and119872119860

Ωas those of 119879119860

Ω120572and 119879119860

Ω

Before stating our main results we introduce somedefinitions and notations at first

A weight is a locally integrable function on R119899 whichtakes values in (0infin) almost everywhere For a weight 119908

and a measurable set 119864 we define 119908(119864) = int119864119908(119909)119889119909 the

Lebesgue measure of 119864 by |119864| and the characteristic functionof 119864 by 120594

119864 The weighted Lebesgue spaces with respect to the

measure 119908(119909)119889119909 are denoted by 119871119901(119908) with 0 lt 119901 lt infinWe say a weight 119908 satisfies the doubling condition if thereexists a constant 119863 gt 0 such that for any ball 119861 we have119908(2119861) le 119863119908(119861) When 119908 satisfies this condition we denote119908 isin Δ

2for short

Throughout this paper 119861(1199090 119903) denotes a ball centered at

1199090with radius 119903 Let 119876 be a cube with sides parallel to the

axes For119870 gt 0119870119876 denotes the cube with the same center as119876 and side-length being119870 times longer When 120572 = 0 we willdenote 119879

Ω120572 119879119860Ω120572

119872119860

Ω120572by 119879

Ω 119879119860Ω119872119860

Ω respectively And for

any number 119886 1198861015840 stands for the conjugate of 119886 The letter 119862denotes a positive constant that may vary at each occurrencebut is independent of the essential variable

Next we give the definition of weighted Morrey spaceintroduced in [10]

Definition 1 Let 1 le 119901 lt infin let 0 lt 120581 lt 1 and let 119908 be aweight Then the weighted Morrey space is defined by

119871119901120581(119908) fl 119891 isin 119871119901loc (119908)

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908) lt infin (7)

where

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908) = sup

119861

(1

119908 (119861)120581int119861

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119901119908 (119909) 119889119909)

1119901

(8)


When we investigate the boundedness of the multilinearfractional integral operator we need to consider the weightedMorrey space with two weights It is defined as follows

Definition 2 Let 1 le 119901 lt infin let 0 lt 120581 lt 1 and let 119906 V be twoweights The two weights weighted Morrey space is definedby

119871119901120581(119906 V) fl 119891 1003817100381710038171003817119891

1003817100381710038171003817119871119901120581(119906V) lt infin (9)

where

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119906V) = sup

119861

(1

V (119861)120581int119861

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119901119906 (119909) 119889119909)

1119901

(10)

and the supremum is taken over all balls 119861 in R119899 If 119906 = Vthen we denote 119871119901120581(119906) for short

As is pointed out in [10] we could also define theweightedMorrey spaces with cubes instead of balls So we shall usethese two definitions of weighted Morrey spaces appropriateto calculation

Then we give the definitions of Lipschitz space and 119861119872119874space

Definition 3 The Lipschitz space of order 120573 0 lt 120573 lt 1 isdefined by

Λ120573(R119899) = 119891

1003816100381610038161003816119891 (119909) minus 119891 (119910)1003816100381610038161003816 le 119862

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

120573 (11)

and the smallest constant 119862 gt 0 is the Lipschitz norm sdot Λ120573



119887lowast = 119887BMO = sup119861

1

|119861|int119861

1003816100381610038161003816119887 (119909) minus 1198871198611003816100381610038161003816 119889119909 lt infin (12)

where

119887119861=1

|119861|int119861

119887 (119910) 119889119910 (13)





for any ball 119861

(1

|119861|int119861

119908 (119909) 119889119909)(1

|119861|int119861

119908 (119909)minus1(119901minus1)

119889119909)

119901minus1

le 119862 (14)


1lt119901ltinfin119860119901



119872119908(119909) le 119862119908 (119909) (15)


(1

|119861|int119861

119908 (119909)119902119889119909)

1119902

sdot (1

|119861|int119861

119908 (119909)minus119901(119901minus1)

119889119909)

(119901minus1)119901

le 119862

(16)


(1

|119861|int119861

119908 (119909)119902119889119909)

1119902


1

119908 (119909)) le 119862 (17)


(a) 119908119902 119908minus1199011015840

119908minus1199021015840

isin Δ2

(b) 119908minus1199011015840

isin 1198601199051015840 with 119905 = 1 + 1199021199011015840




1119902 = 1119901 minus (120572 + 120573)119899 0 lt 120581 lt 119901119902 1199081199041015840

isin 119860(1199011199041015840 119902119904

1015840)

119863120574119860 isin Λ

120573(|120574| = 119898 minus 1) then

10038171003817100381710038171003817119879119860

Ω12057211989110038171003817100381710038171003817119871119902120581119902119901(119908119902)

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) (18)

10038171003817100381710038171003817119872119860

Ω12057211989110038171003817100381710038171003817119871119902120581119902119901(119908119902)

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) (19)



1119901 minus120573119899 0 lt 120581 lt 1199011199021199081199041015840

isin 119860(1199011199041015840 119902119904

1015840)119863120574119860 isin Λ

120573(|120574| =


10038171003817100381710038171003817119879119860

Ω11989110038171003817100381710038171003817119871119902120581119902119901(119908119902)

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) (20)

10038171003817100381710038171003817119872119860

Ω11989110038171003817100381710038171003817119871119902120581119902119901(119908119902)

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) (21)



1015840

isin 119860(1199011199041015840 119902119904

1015840) 119863120574119860 isin BMO (|120574| = 119898 minus 1)

then

10038171003817100381710038171003817119879119860

Ω12057211989110038171003817100381710038171003817119871119902120581119902119901(119908119902)

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) (22)

10038171003817100381710038171003817119872119860

Ω12057211989110038171003817100381710038171003817119871119902120581119902119901(119908119902)

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) (23)


Ωby [119860 119879

Ω]

[119860119872Ω] and

119860

Ω 119860


from119879119860Ωand119872119860


precise

[119860 119879Ω] 119891 (119909) = int

R119899

Ω(119909 minus 119910)

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899(119860 (119909) minus 119860 (119910))

sdot 119891 (119910) 119889119910

[119860119872Ω] 119891 (119909) = sup

119903gt0

1

119903119899int|119909minus119910|lt119903

Ω(119909 minus 119910)

sdot (119860 (119909) minus 119860 (119910)) 119891 (119910) 119889119910

119860

Ω119891 (119909) = int

R119899

Ω(119909 minus 119910)

1003816100381610038161003816119909 minus 1199101003816100381610038161003816


sdot (119909 minus 119910)) 119891 (119910) 119889119910

119860

Ω119891 (119909) = sup

119903gt0

1

119903119899+1int|119909minus119910|lt119903

Ω(119909 minus 119910)


(24)



119878119899minus1Ω(119909

1015840)119889120590(119909

1015840) =


1003817100381710038171003817[119860 119879Ω] 1198911003817100381710038171003817119871119901120581(119908) le 119862 119860lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908) (25)

1003817100381710038171003817[119860119872Ω] 1198911003817100381710038171003817119871119901120581(119908) le 119862 119860lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908) (26)



119878119899minus1120579Ω(120579)119889120579 = 0 1 lt 119901 lt


100381710038171003817100381710038171003817119860

Ω119891100381710038171003817100381710038171003817119871119901120581(119908)

le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908) (27)

100381710038171003817100381710038171003817119860

Ω119891100381710038171003817100381710038171003817119871119901120581(119908)

le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908) (28)


Ω when



Remark 14 Define

1198791198601119860119896

Ω120572119891 (119909) = int

R119899

119896

prod

119894=1

119877119898119894

(119860119894 119909 119910)

sdotΩ (119909 minus 119910)

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572+119873119891 (119910) 119889119910

1198721198601119860119896

Ω120572119891 (119909) = sup

119903gt0

1


1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

sdot

119896

prod

119894=1

10038161003816100381610038161003816119877119898119894

(119860119894 119909 119910)

10038161003816100381610038161003816

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

(29)

where 119877119898119894

(119860119894 119909 119910) = 119860

119894(119909) minus sum

|120574|lt119898119894

(1120574)119863120574119860119894(119910)(119909 minus

119910)120574 119894 = 1 119896 119873 = sum

119896

119894=1(119898119894minus 1) When 0 lt 120572 lt 119899


Ω120572and 119872119860

1119860119896



119894minus1119863120574119860119894Λ120573


119894=1(sum|120574|=119898


hold respectively




119899) for some 119897 gt 119899 Then

1003816100381610038161003816119877119898 (119860 119909 119910)1003816100381610038161003816

le 1198621003816100381610038161003816119909 minus 119910

1003816100381610038161003816

119898sum

|120574|=119898

(1

1003816100381610038161003816119868119910

119909

1003816100381610038161003816

int119868119910

119909

1003816100381610038161003816119863120574119860 (119911)

1003816100381610038161003816

119897119889119911)

1119897

(30)




10038171003817100381710038171198911003817100381710038171003817Λ120573

asymp sup119876

1

|119876|1+120573119899

int119876

1003816100381610038161003816119891 (119909) minus 119887119876 (119891)1003816100381610038161003816 119889119909

asymp sup119876

1

|119876|120573119899(1

|119876|int119876

1003816100381610038161003816119891 (119909) minus 119887119876 (119891)1003816100381610038161003816

119902119889119909)

1119902

(31)


Lemma 17 (see [13]) Let 1198761sub 119876

2 119892 isin Λ

120573(0 lt 120573 lt 1) Then

100381610038161003816100381610038161198921198761

minus 1198921198762

10038161003816100381610038161003816le 119862

100381610038161003816100381611987621003816100381610038161003816

120573119899 10038171003817100381710038171198921003817100381710038171003817Λ120573

(32)



119901(119908119901) to


conditions

(a) 1199041015840 lt 119901 and 119908(119909)1199041015840

isin 119860(1199011199041015840 119902119904

1015840)

(b) 119904 gt 119902 and 119908(119909)minus1199041015840

isin 119860(11990210158401199041015840 11990110158401199041015840)


such that 119908(119909)1199031015840

isin 119860(119901 119902)



119908 (2119861) ge 1198631119908 (119861) (33)





119894ge 2 and



(a) 1199041015840 lt 119901 and 119908(119909)1199041015840

isin 119860(1199011199041015840 119902119904

1015840)

(b) 119904 gt 119902 and 119908(119909)minus1199041015840

isin 119860(11990210158401199041015840 11990110158401199041015840)


1015840

isin 119860(119901 119902)


(intR119899

100381610038161003816100381610038161198791198601119860119896

Ω120572119891 (119909)119908 (119909)

10038161003816100381610038161003816

119902

119889119909)

1119902

le 119862

119896

prod

119894=1

( sum

|120574|=119898119894minus1

1003817100381710038171003817119863120574119860119894

1003817100381710038171003817lowast)

sdot (intR119899

1003816100381610038161003816119891 (119909)119908 (119909)1003816100381610038161003816

119901119889119909)

1119901

(34)


1

|119876|int119876

1003816100381610038161003816119887 minus 1198871198761003816100381610038161003816

119901119889119909 le 119862 119887

119901

lowast (35)



2

100381610038161003816100381610038161198871198761

minus 1198871198762

10038161003816100381610038161003816le 119862 log(

100381610038161003816100381611987621003816100381610038161003816

100381610038161003816100381611987611003816100381610038161003816

) 119887lowast (36)


1003816100381610038161003816 le 2119899(119895 + 1) 119887lowast (37)



119878119899minus1Ω(119909

1015840)119889120590(119909

1015840) = 0 If 119887 isin BMO(R119899) then [119887 119879

Ω]




isin 11986011990110158401199041015840

(c) 1 le 119901 lt infin and 1199081199041015840

isin 119860119901




Ω119891100381710038171003817100381710038171003817119871119901(119908)




2 More


119908 (120582119876) le 119862120582119899119901119908 (119876) (39)



119908 (119878)

119908 (119876)le 119862(

|119878|

|119876|)

120575

(40)



BMO (119908) = 119887 119887lowast119908

= sup119876

1

119908 (119876)int119876

1003816100381610038161003816119887 (119909) minus 1198981198761199081198871003816100381610038161003816 119908 (119909) 119889119909

119898119876119908119887 =

1

119908 (119876)int119876

119887 (119909)119908 (119909) 119889119909

(41)



Ω120572119891(119909) at first Set

119879Ω120572+120573

119891 (119909) = intR119899

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572minus120573

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

0 lt 120572 + 120573 lt 119899

(42)




10038161003816100381610038161003816119879119860

Ω120572119891 (119909)

10038161003816100381610038161003816le 119862( sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573

)119879Ω120572+120573

119891 (119909) (43)


119896= 2

119896119876 and set

119860119876119896

(119910) = 119860 (119910) minus sum

|120574|=119898minus1

1

120574119898119876119896

(119863120574119860)119910

120574 (44)

where 119898119876119896


|120574| = 119898 minus 1

119863120574119860119876119896

(119910) = 119863120574119860 (119910) minus 119898

119876119896

(119863120574119860) (45)


119877119898(119860 119909 119910) = 119877

119898(119860119876119896

119909 119910) (46)

Hence

10038161003816100381610038161003816119879119860

Ω120572119891 (119909)

10038161003816100381610038161003816

le

infin

sum

119896=minusinfin


119896119903

10038161003816100381610038161003816119877119898(119860119876119896

119909 119910)10038161003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119898minus1

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

flinfin

sum

119896=minusinfin

119879119896

(47)

By Lemma 15 we get

10038161003816100381610038161003816119877119898(119860119876119896

119909 119910)10038161003816100381610038161003816

le10038161003816100381610038161003816119877119898minus1

(119860119876119896

119909 119910)10038161003816100381610038161003816

+ 119862 sum

|120574|=119898minus1

10038161003816100381610038161003816119863120574119860119876119896

(119910)10038161003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119898minus1

le 1198621003816100381610038161003816119909 minus 119910

1003816100381610038161003816

119898minus1sum

|120574|=119898minus1

(1

1003816100381610038161003816119868119910

119909

1003816100381610038161003816

int119868119910

119909

10038161003816100381610038161003816119863120574119860119876119896

(119911)10038161003816100381610038161003816

119897

119889119911)

1119897

+ 1198621003816100381610038161003816119909 minus 119910

1003816100381610038161003816

119898minus1sum

|120574|=119898minus1

10038161003816100381610038161003816119863120574119860119876119896

(119910)10038161003816100381610038161003816

(48)



119896 By Lemmas 16


(1

1003816100381610038161003816119868119910

119909

1003816100381610038161003816

int119868119910

119909

10038161003816100381610038161003816119863120574119860119876119896

(119911)10038161003816100381610038161003816

119897

119889119911)

1119897

= (1

1003816100381610038161003816119868119910

119909

1003816100381610038161003816

int119868119910

119909

10038161003816100381610038161003816119863120574119860 (119911) minus 119898119876

119896

(119863120574119860)10038161003816100381610038161003816

119897

119889119911)

1119897

le (1

1003816100381610038161003816119868119910

119909

1003816100381610038161003816

int119868119910

119909

10038161003816100381610038161003816119863120574119860 (119911) minus 119898

119868119910

119909

(119863120574119860)10038161003816100381610038161003816

119897

119889119911)

1119897

+10038161003816100381610038161003816119898119868119910

119909

(119863120574119860) minus 119898

5119899119876119896

(119863120574119860)10038161003816100381610038161003816

+100381610038161003816100381610038161198985119899119876119896

(119863120574119860) minus 119898

119876119896

(119863120574119860)10038161003816100381610038161003816

le 11986210038161003816100381610038161198761198961003816100381610038161003816

120573119899 10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573

le 119862 (2119896119903)120573 1003817100381710038171003817119863

1205741198601003817100381710038171003817Λ120573

(49)


10038161003816100381610038161003816119863120574119860119876119896

(119910)10038161003816100381610038161003816=10038161003816100381610038161003816119863120574119860 (119910) minus 119898

119876119896

(119863120574119860)10038161003816100381610038161003816

le 11986210038161003816100381610038161198761198961003816100381610038161003816

120573119899 10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573

le 119862 (2119896119903)120573 1003817100381710038171003817119863

1205741198601003817100381710038171003817Λ120573

(50)

Thus

10038161003816100381610038161003816119877119898(119860119876119896

119909 119910)10038161003816100381610038161003816

le 1198621003816100381610038161003816119909 minus 119910

1003816100381610038161003816

119898minus1(2119896119903)120573

sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573

(51)

Therefore

119879119896le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573


119896119903

(2119896119903)120573

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572

1003816100381610038161003816Ω (119909 minus 119910)10038161003816100381610038161003816100381610038161003816119891 (119910)

1003816100381610038161003816 119889119910

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573


119896119903

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572minus120573

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

(52)

It follows that

10038161003816100381610038161003816119879119860

Ω120572119891 (119909)

10038161003816100381610038161003816le

infin

sum

119896=minusinfin

(119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573


119896119903

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572minus120573

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910)

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573

sdot

infin

sum

119896=minusinfin


119896119903

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572minus120573

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

= 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573

intR119899

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572minus120573

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

= 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573

119879Ω120572+120573

119891 (119909)

(53)






1+ 119891

2with

1198911= 119891120594


(1

119908119902 (119861)120581119902119901

int119861

10038161003816100381610038161003816119879Ω120572+120573

119891 (119909)10038161003816100381610038161003816

119902

119908119902(119909) 119889119909)

1119902

le1

119908119902 (119861)120581119901(int119861

10038161003816100381610038161003816119879Ω120572+120573

1198911 (119909)

10038161003816100381610038161003816

119902

119908119902(119909) 119889119909)

1119902

+1

119908119902 (119861)120581119901(int119861

10038161003816100381610038161003816119879Ω120572+120573

1198912 (119909)

10038161003816100381610038161003816

119902

119908119902(119909) 119889119909)

1119902

fl 1198691+ 1198692

(54)


2


1198691le

1

119908119902 (119861)120581119901

10038171003817100381710038171003817119879Ω120572+120573

1198911

10038171003817100381710038171003817119871119902(119908119902)

le119862

119908119902 (119861)120581119901

100381710038171003817100381711989111003817100381710038171003817119871119901(119908119901)

=119862

119908119902 (119861)120581119901(int2119861

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119901119908 (119909)

119901119889119909)

1119901

le 11986210038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

119908119902(2119861)

120581119901

119908119902 (119861)120581119901

le 11986210038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

(55)



10038161003816100381610038161003816119879Ω120572+120573

1198912 (119909)

10038161003816100381610038161003816=

infin

sum

119895=1

int2119895+11198612119895119861

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572minus120573

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

le 119862

infin

sum

119895=1

(int2119895+1119861

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

119904119889119910)

1119904

sdot (int2119895+11198612119895119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816

1199041015840

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

(119899minus120572minus120573)1199041015840119889119910)

11199041015840

fl 119862infin

sum

119895=1

(11986811198951198682119895)

(56)



1198681119895= (int

2119895+2119861

|Ω (119911)|119904119889119911)

1119904

= (int

2119895+2

119903119861

0

int119878119899minus1

10038161003816100381610038161003816Ω (119911

1015840)10038161003816100381610038161003816

119904

1198891199111015840119903119899minus1119889119903)

1119904

= 119862 Ω119871119904(119878119899minus1)

100381610038161003816100381610038162119895+211986110038161003816100381610038161003816

1119904

(57)


1198682119895le

119862

10038161003816100381610038162119895+1119861

1003816100381610038161003816

1minus(120572+120573)119899(int2119895+1119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816

1199041015840

119889119910)

11199041015840

(58)


isin 119860(1199011199041015840 119902119904

1015840) we get

(int2119895+1119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816

1199041015840

119889119910)

11199041015840

le 119862(int2119895+1119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119901119908 (119910)

119901119889119910)

1119901

sdot (int2119895+1119861

119908 (119910)minus1199011199041015840

(119901minus1199041015840

)119889119910)

(119901minus1199041015840

)1199011199041015840

le 11986210038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) 119908

119902(2119895+1119861)120581119901

sdot (int2119895+1119861

119908 (119910)minus1199011199041015840

(119901minus1199041015840

)119889119910)

(119901minus1199041015840

)1199011199041015840

le 11986210038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) 119908

119902(2119895+1119861)120581119901

sdot

100381610038161003816100381610038162119895+111986110038161003816100381610038161003816

(119901119902minus1199041015840

119902+1199041015840

119901)1199011199021199041015840

119908119902 (2119895+1119861)1119902

(59)

Thus

10038161003816100381610038161003816119879Ω120572+120573

1198912 (119909)

10038161003816100381610038161003816le 119862

infin

sum

119895=1

(11986811198951198682119895)

le 119862

infin

sum

119895=1

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

1

119908119902 (2119895+1119861)1119902minus120581119901

(60)

So we get

1198692le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

infin

sum

119895=1

119908119902(119861)

1119902minus120581119901

119908119902 (2119895+1119861)1119902minus120581119901

(61)


1198692le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) (62)



Ω120572is bounded




119879119860

Ω120572119891 (119909)

= intR119899

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572+119898minus1

1003816100381610038161003816119877119898 (119860 119909 119910)10038161003816100381610038161003816100381610038161003816119891 (119910)

1003816100381610038161003816 119889119910

0 le 120572 lt 119899

(63)


119860

Ω120572 On the


119879119860

Ω120572119891 (119909)

ge int|119909minus119910|lt119903

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572+119898minus1

1003816100381610038161003816119877119898 (119860 119909 119910)10038161003816100381610038161003816100381610038161003816119891 (119910)

1003816100381610038161003816 119889119910

ge1

119903119899minus120572+119898minus1


1003816100381610038161003816Ω (119909 minus 119910)10038161003816100381610038161003816100381610038161003816119877119898 (119860 119909 119910)

10038161003816100381610038161003816100381610038161003816119891 (119910)

1003816100381610038161003816 119889119910

(64)


119879119860

Ω120572119891 (119909) ge 119872

119860

Ω120572119891 (119909) (65)




119879Ω120573119891 (119909) = int

R119899

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120573

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910 (66)







119860119876(119910) = 119860 (119910) minus sum

|120574|=119898minus1

1

120574119898119876(119863120574119860)119910

120574 (67)


119876+ 119891120594

(119876)119888 = 119891

1+ 119891

2

Then we have

1

119908119902 (119876)120581119901(int119876

10038161003816100381610038161003816119879119860

Ω120572119891 (119910)

10038161003816100381610038161003816

119902

119908 (119910)119902119889119910)

1119902

le1

119908119902 (119876)120581119901(int119876

10038161003816100381610038161003816119879119860

Ω1205721198911(119910)10038161003816100381610038161003816

119902

119908 (119910)119902119889119910)

1119902

+1

119908119902 (119876)120581119901(int119876

10038161003816100381610038161003816119879119860

Ω1205721198912(119910)10038161003816100381610038161003816

119902

119908 (119910)119902119889119910)

1119902

fl 119868 + 119868119868

(68)


119868 le119862

119908119902 (119876)120581119901

sdot sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast (int

119876

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119901119908 (119910)

119901119889119910)

1119901

= 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)(

119908119902(119876)

119908119902 (119876))

120581119901

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

(69)


Ω1205721198912(119910) contained in 119868119868 By


10038161003816100381610038161003816119879119860

Ω1205721198912(119910)10038161003816100381610038161003816

le int(119876)119888

10038161003816100381610038161003816119877119898(119860119876 119910 119911)

100381610038161003816100381610038161003816100381610038161003816119910 minus 119911

1003816100381610038161003816

119898minus1

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

le 119862int(119876)119888

sum

|120574|=119898minus1

(1

10038161003816100381610038161003816119868119911119910

10038161003816100381610038161003816

int119868119911

119910

10038161003816100381610038161003816119863120574119860119876(119905)10038161003816100381610038161003816

119897

119889119905)

1119897

sdot1003816100381610038161003816Ω (119910 minus 119911)

1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572119889119911

+ 119862int(119876)119888

sum

|120574|=119898minus1

10038161003816100381610038161003816119863120574119860 (119911) minus 119898

119876(119863120574119860)10038161003816100381610038161003816

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

sdot

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572119889119911 fl 119868119868

1+ 119868119868

2

(70)

We estimate 1198681198681and 119868119868



isin 119860(1199011199041015840 119902119904

1015840) we get

1198681198681le 119862 sum

|120574|=119898minus1

(1

10038161003816100381610038161003816119868119911119910

10038161003816100381610038161003816

int119868119911

119910

10038161003816100381610038161003816119863120574119860 (119905) minus 119898

119876(119863120574119860)10038161003816100381610038161003816

119897

119889119905)

1119897

sdot

infin

sum

119895=1

int2119895+11198762119895119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572119889119911

le 119862 sum

|120574|=119898minus1

[

[

(1

10038161003816100381610038161003816119868119911119910

10038161003816100381610038161003816

sdot int119868119911

119910

100381610038161003816100381610038161003816119863120574119860 (119905) minus 119898119868119911

119910

(119863120574119860)100381610038161003816100381610038161003816

119897

119889119905)

1119897

+100381610038161003816100381610038161003816119898119868119911

119910

(119863120574119860)

minus 1198985119899119876(119863120574119860)100381610038161003816100381610038161003816+100381610038161003816100381610038161198985119899119876(119863120574119860) minus 119898

119876(119863120574119860)10038161003816100381610038161003816]

]

sdot

infin

sum

119895=1

(int2119895+1119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

119904119889119911)

1119904

sdot (int2119895+11198762119895119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

(119899minus120572)1199041015840119889119911)

11199041015840

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast Ω119871

119904(119878119899minus1)

sdot

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus120572119899(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840


le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus120572119899(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

119901

sdot 119908 (119911)119901119889119911)

1119901

sdot (int2119895+1119876

119908 (119911)minus1199011199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

sdot

infin

sum

119895=1

1

119908119902 (2119895+1119876)1119902minus120581119901

(71)


1198681198682le 119862

infin

sum

119895=1

int2119895+11198762119895119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

sdot1003816100381610038161003816Ω (119910 minus 119911)

1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572119889119911

+ 119862

infin

sum

119895=1

int2119895+11198762119895119876

sum

|120574|=119898minus1

100381610038161003816100381610038161198982119895+1119876(119863120574119860) minus 119898

119876(119863120574119860)10038161003816100381610038161003816

sdot1003816100381610038161003816Ω (119910 minus 119911)

1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572119889119911 fl 119868119868

21+ 119868119868

22

(72)


11986811986821le 119862

infin

sum

119895=1

1

100381610038161003816100381621198951198761003816100381610038161003816

1minus

120572

119899

int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863

120574119860)10038161003816100381610038161003816100381610038161003816Ω (119910 minus 119911)

10038161003816100381610038161003816100381610038161003816119891 (119911)

1003816100381610038161003816 119889119911

le 119862

infin

sum

119895=1

1

100381610038161003816100381621198951198761003816100381610038161003816

1minus

120572

119899

(int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

1199041015840

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840

(int2119895+1119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

119904119889119911)

1119904

le 119862 Ω119871119904(119878119899minus1)

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus

120572

119899

(int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

1199041015840

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840

le 119862 Ω119871119904(119878119899minus1)

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus

120572

119899

(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

119901119908 (119911)

119901119889119911)

1119901

sdot (int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

1199011199041015840

(119901minus1199041015840

)119908 (119911)

minus1199011199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

(73)


(int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

1199011199041015840

(119901minus1199041015840

)119908 (119911)

minus1199011199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

le 119862(int2119895+1119876

sum

|120574|=119898minus1

10038161003816100381610038161003816119863120574119860 (119911) minus 119898

2119895+1119876119908minus1199011199041015840(119901minus1199041015840)(119863120574119860)10038161003816100381610038161003816

1199011199041015840

(119901minus1199041015840

)

119908 (119911)minus1199011199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

+ sum

|120574|=119898minus1


2119895+1119876(119863120574119860)10038161003816100381610038161003816119908minus1199011199041015840

(119901minus1199041015840

)(2119895+1119876)(119901minus1199041015840

)1199011199041015840

fl 119868119868119868 + 119868119881

(74)


isin 119860(1199011199041015840 119902119904


119908minus1199011199041015840

(119901minus1199041015840

)isin 119860

1199051015840 sub 119860



(119901minus1199041015840


of BMO(R119899) and 1199081199041015840

isin 119860(1199011199041015840 119902119904

1015840) we have


119868119868119868 le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast 119908

minus1199011199041015840

(119901minus1199041015840

)(2119895+1119876)(119901minus1199041015840

)1199011199041015840

= 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

100381610038161003816100381610038162119895+111987610038161003816100381610038161003816

(119901minus1199041015840

)1199011199041015840

+1119902

119908119902 (2119895+1119876)1119902

(75)



119905 isin 2119895+1119876 sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816 gt 119904

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le 1198621

100381610038161003816100381610038162119895+111987610038161003816100381610038161003816119890minus1198622119904(sum|120574|=119898minus1

119863120574

119860lowast)

(76)



119908(

119905 isin 2119895+1119876 sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

gt 119904

) le 119862119908(2119895+1119876) 119890

minus1198622119904120575(sum

|120574|=119898minus1119863120574

119860lowast)

(77)


sum

|120574|=119898minus1


2119895+1119876(119863120574119860)10038161003816100381610038161003816

le1

119908minus1199011199041015840(119901minus119904

1015840) (2119895+1119876)

int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119905)

minus 1198982119895+1119876(119863120574119860)1003816100381610038161003816 119908minus1199011199041015840

(119901minus1199041015840

)(119905) 119889119905

=119862

119908minus1199011199041015840(119901minus119904

1015840) (2119895+1119876)

int

infin

0

119908minus1199011199041015840

(119901minus1199041015840

)(

119905

isin 2119895+1119876 sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

gt 119904

)119889119904 le119862

119908minus1199011199041015840(119901minus119904

1015840) (2119895+1119876)

sdot int

infin

0

119908minus1199011199041015840

(119901minus1199041015840

)(2119895+1119876) 119890

minus1198622119904120575(sum

|120574|=119898minus1119863120574

119860lowast)119889119904

= 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

(78)

As a result

119868119881 le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast 119908

minus1199011199041015840

(119901minus1199041015840

)(2119895+1119876)(119901minus1199041015840

)1199011199041015840

= 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

100381610038161003816100381610038162119895+111987610038161003816100381610038161003816

(119901minus1199041015840

)1199011199041015840

+1119902

119908119902 (2119895+1119876)1119902

(79)

Thus

11986811986821le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

sdot

infin

sum

119895=1

1

119908119902 (2119895+1119876)1119902minus120581119901

(80)


1199081199041015840

isin 119860(1199011199041015840 119902119904

1015840) we get

11986811986822le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

sdot

infin

sum

119895=1

119895 int2119895+11198762119895119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572119889119911

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast Ω119871

119904(119878119899minus1)

infin

sum

119895=1

119895

sdot

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus120572119899(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

infin

sum

119895=1

119895

sdot

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus120572119899(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

119901119908 (119911)

119901119889119911)

1119901

sdot (int2119895+1119876

119908 (119911)minus1199011199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

sdot

infin

sum

119895=1

119895

119908119902 (2119895+1119876)1119902minus120581119901

(81)

Hence

10038161003816100381610038161003816119879119860

Ω1205721198912(119910)10038161003816100381610038161003816le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

sdot

infin

sum

119895=1

119895

119908119902 (2119895+1119876)1119902minus120581119901

(82)

Therefore

119868119868 le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

sdot

infin

sum

119895=1

119895119908119902(119876)

1119902minus120581119901

119908119902 (2119895+1119876)1119902minus120581119901


le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

infin

sum

119895=1

119895

(119863119895+1

1)1119902minus120581119901

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

(83)


1

119908119902 (119876)120581119901(int119876

10038161003816100381610038161003816119879119860

Ω120572119891 (119910)

10038161003816100381610038161003816

119902

119908 (119910)119902119889119910)

1119902

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

(84)




119876+ 119891120594

(119876)119888 fl 119891

1+ 119891

2 Then we have

1

119908 (119876)120581119901(int119876

1003816100381610038161003816[119860 119879Ω] 119891 (119910)1003816100381610038161003816

119901119908 (119910) 119889119910)

1119901

le1

119908 (119876)120581119901(int119876

1003816100381610038161003816[119860 119879Ω] 1198911 (119910)1003816100381610038161003816

119901119908 (119910) 119889119910)

1119901

+1

119908 (119876)120581119901(int119876

1003816100381610038161003816[119860 119879Ω] 1198912 (119910)1003816100381610038161003816

119901119908 (119910) 119889119910)

1119901

fl 119868 + 119868119868

(85)


119868 le1

119908 (119876)120581119901

1003817100381710038171003817[119860 119879Ω] 11989111003817100381710038171003817119871119901(119908)

le119862

119908 (119876)120581119901119860lowast

100381710038171003817100381711989111003817100381710038171003817119871119901(119908)

= 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

119908(119876)120581119901

119908 (119876)120581119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(86)


1003816100381610038161003816[119860 119879Ω] 1198912 (119910)1003816100381610038161003816 le

infin

sum

119895=1

int2119895+11198762119895119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899

1003816100381610038161003816119860 (119910) minus 119860 (119911)1003816100381610038161003816

sdot1003816100381610038161003816119891 (119911)

1003816100381610038161003816 119889119911

le 119862

infin

sum

119895=1

1

100381610038161003816100381621198951198761003816100381610038161003816

(int2119895+1119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

119904119889119911)

1119904

sdot (int2119895+1119876

1003816100381610038161003816119860 (119910) minus 119860 (119911)1003816100381610038161003816

1199041015840

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840

le 119862 Ω119871119904

sdot

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

sdot (int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840

+ 119862 Ω119871119904

sdot

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

(int2119895+1119876


10038161003816100381610038161003816

1199041015840

sdot1003816100381610038161003816119891 (119911)

1003816100381610038161003816

1199041015840

119889119911)

11199041015840

fl 1198681198681(119910) + 119868119868

2

(87)




1

119908 (119876)120581119901(int119876

1198681198681(119910)

119901119908 (119910) 119889119910)

1119901

= 119862Ω119871119904

119908 (119876)120581119901

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

(int119876

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

119901

(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

1199011199041015840

119908 (119910) 119889119910)

1119901

le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

119908 (119876)

120581

119901

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

119908 (2119895+1119876)120581119901

119908minus1199041015840

(119901minus1199041015840

)(2119895+1119876)(119901minus1199041015840

)1199011199041015840

sdot (int119876

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

119908 (119876)120581119901

infin

sum

119895=1

1

119908 (2119895+1119876)(1minus120581)119901

(int119876

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

(88)



(int119876

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

le (int119876

1003816100381610038161003816119860 (119910) minus 119898119876119908 (119860)1003816100381610038161003816

119901119908 (119910) 119889119910)

1119901

+10038161003816100381610038161003816119898119876119908 (119860) minus 1198982119895+1119876119908minus119904

1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816119908 (119876)

1119901

fl 119868119868119868 + 119868119881

(89)



Lemma 25 that

119868119868119868 le 119860lowast 119908 (119876)1119901 (90)


10038161003816100381610038161003816119898119876119908 (119860) minus 1198982119895+1119876119908minus119904

1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

le1003816100381610038161003816119898119876119908 (119860) minus 119898119876 (119860)

1003816100381610038161003816 +1003816100381610038161003816119898119876 (119860) minus 1198982119895+1119876 (119860)

1003816100381610038161003816

+100381610038161003816100381610038161198982119895+1119876 (119860) minus 1198982119895+1119876119908minus119904

1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816le

1

119908 (119876)

sdot int119876

1003816100381610038161003816119860 (119905) minus 119898119876 (119860)1003816100381610038161003816 119908 (119905) 119889119905 + 2

119899(119895 + 1) 119860lowast

+1

119908minus1199041015840(119901minus119904

1015840) (2119895+1119876)

sdot int2119895+1119876

1003816100381610038161003816119860 (119905) minus 1198982119895+1119876 (119860)1003816100381610038161003816 119908minus1199041015840

(119901minus1199041015840

)(119905) 119889119905

le 119862 (119895 + 1) 119860lowast

(91)

Hence

119868119881 le 119862 (119895 + 1) 119860lowast 119908 (119876)1119901 (92)

As a result

1

119908 (119876)120581119901(int119876

1198681198681(119910)

119901119908 (119910) 119889119910)

1119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

infin

sum

119895=1

(119895 + 1)119908 (119876)

(1minus120581)119901

119908 (2119895+1119876)(1minus120581)119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(93)


1199011199041015840 we get

1198681198682le 119862 Ω119871119904

sdot

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

119901119908 (119911) 119889119911)

1119901

sdot (int2119895+1119876


10038161003816100381610038161003816

1199011199041015840

(119901minus1199041015840

)

sdot 119908 (119911)minus1199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

le 119862 Ω119871119904 119860lowast

sdot10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

119908 (2119895+1119876)120581119901

sdot 119908minus1199041015840

(119901minus1199041015840

)(2119895+1119876)(119901minus1199041015840

)1199011199041015840

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

sdot

infin

sum

119895=1

1

119908 (2119895+1119876)(1minus120581)119901

(94)

Therefore

1

119908 (119876)120581119901(int119876

119868119868119901

2119908 (119910) 119889119910)

1119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

infin

sum

119895=1

119908 (119876)(1minus120581)119901

119908 (2119895+1119876)(1minus120581)119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(95)




119860119876(119910) = 119860 (119910) minus 119898

119876(nabla119860) 119910 (96)


(119876)119888 fl

1198911+ 119891

2 Then we get

1

119908 (119876)120581119901(int119876

100381610038161003816100381610038161003816119860

Ω119891 (119910)

100381610038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

le1

119908 (119876)120581119901(int119876

100381610038161003816100381610038161003816119860

Ω1198911(119910)100381610038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

+1

119908 (119876)120581119901(int119876

100381610038161003816100381610038161003816119860

Ω1198912(119910)100381610038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

fl 119868 + 119868119868

(97)



119868 le1

119908 (119876)120581119901

100381710038171003817100381710038171003817119860

Ω1198911

100381710038171003817100381710038171003817119871119901(119908)

le119862


100381710038171003817100381711989111003817100381710038171003817119871119901(119908)

le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

119908(119876)120581119901

119908 (119876)120581119901

le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(98)


119901 Ω isin 119871




Competing Interests


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119887lowast = 119887BMO = sup119861

1

|119861|int119861

1003816100381610038161003816119887 (119909) minus 1198871198611003816100381610038161003816 119889119909 lt infin (12)

where

119887119861=1

|119861|int119861

119887 (119910) 119889119910 (13)





for any ball 119861

(1

|119861|int119861

119908 (119909) 119889119909)(1

|119861|int119861

119908 (119909)minus1(119901minus1)

119889119909)

119901minus1

le 119862 (14)


1lt119901ltinfin119860119901



119872119908(119909) le 119862119908 (119909) (15)


(1

|119861|int119861

119908 (119909)119902119889119909)

1119902

sdot (1

|119861|int119861

119908 (119909)minus119901(119901minus1)

119889119909)

(119901minus1)119901

le 119862

(16)


(1

|119861|int119861

119908 (119909)119902119889119909)

1119902


1

119908 (119909)) le 119862 (17)


(a) 119908119902 119908minus1199011015840

119908minus1199021015840

isin Δ2

(b) 119908minus1199011015840

isin 1198601199051015840 with 119905 = 1 + 1199021199011015840




1119902 = 1119901 minus (120572 + 120573)119899 0 lt 120581 lt 119901119902 1199081199041015840

isin 119860(1199011199041015840 119902119904

1015840)

119863120574119860 isin Λ

120573(|120574| = 119898 minus 1) then

10038171003817100381710038171003817119879119860

Ω12057211989110038171003817100381710038171003817119871119902120581119902119901(119908119902)

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) (18)

10038171003817100381710038171003817119872119860

Ω12057211989110038171003817100381710038171003817119871119902120581119902119901(119908119902)

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) (19)



1119901 minus120573119899 0 lt 120581 lt 1199011199021199081199041015840

isin 119860(1199011199041015840 119902119904

1015840)119863120574119860 isin Λ

120573(|120574| =


10038171003817100381710038171003817119879119860

Ω11989110038171003817100381710038171003817119871119902120581119902119901(119908119902)

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) (20)

10038171003817100381710038171003817119872119860

Ω11989110038171003817100381710038171003817119871119902120581119902119901(119908119902)

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) (21)



1015840

isin 119860(1199011199041015840 119902119904

1015840) 119863120574119860 isin BMO (|120574| = 119898 minus 1)

then

10038171003817100381710038171003817119879119860

Ω12057211989110038171003817100381710038171003817119871119902120581119902119901(119908119902)

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) (22)

10038171003817100381710038171003817119872119860

Ω12057211989110038171003817100381710038171003817119871119902120581119902119901(119908119902)

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) (23)


Ωby [119860 119879

Ω]

[119860119872Ω] and

119860

Ω 119860


from119879119860Ωand119872119860


precise

[119860 119879Ω] 119891 (119909) = int

R119899

Ω(119909 minus 119910)

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899(119860 (119909) minus 119860 (119910))

sdot 119891 (119910) 119889119910

[119860119872Ω] 119891 (119909) = sup

119903gt0

1

119903119899int|119909minus119910|lt119903

Ω(119909 minus 119910)

sdot (119860 (119909) minus 119860 (119910)) 119891 (119910) 119889119910

119860

Ω119891 (119909) = int

R119899

Ω(119909 minus 119910)

1003816100381610038161003816119909 minus 1199101003816100381610038161003816


sdot (119909 minus 119910)) 119891 (119910) 119889119910

119860

Ω119891 (119909) = sup

119903gt0

1

119903119899+1int|119909minus119910|lt119903

Ω(119909 minus 119910)


(24)



119878119899minus1Ω(119909

1015840)119889120590(119909

1015840) =


1003817100381710038171003817[119860 119879Ω] 1198911003817100381710038171003817119871119901120581(119908) le 119862 119860lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908) (25)

1003817100381710038171003817[119860119872Ω] 1198911003817100381710038171003817119871119901120581(119908) le 119862 119860lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908) (26)



119878119899minus1120579Ω(120579)119889120579 = 0 1 lt 119901 lt


100381710038171003817100381710038171003817119860

Ω119891100381710038171003817100381710038171003817119871119901120581(119908)

le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908) (27)

100381710038171003817100381710038171003817119860

Ω119891100381710038171003817100381710038171003817119871119901120581(119908)

le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908) (28)


Ω when



Remark 14 Define

1198791198601119860119896

Ω120572119891 (119909) = int

R119899

119896

prod

119894=1

119877119898119894

(119860119894 119909 119910)

sdotΩ (119909 minus 119910)

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572+119873119891 (119910) 119889119910

1198721198601119860119896

Ω120572119891 (119909) = sup

119903gt0

1


1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

sdot

119896

prod

119894=1

10038161003816100381610038161003816119877119898119894

(119860119894 119909 119910)

10038161003816100381610038161003816

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

(29)

where 119877119898119894

(119860119894 119909 119910) = 119860

119894(119909) minus sum

|120574|lt119898119894

(1120574)119863120574119860119894(119910)(119909 minus

119910)120574 119894 = 1 119896 119873 = sum

119896

119894=1(119898119894minus 1) When 0 lt 120572 lt 119899


Ω120572and 119872119860

1119860119896



119894minus1119863120574119860119894Λ120573


119894=1(sum|120574|=119898


hold respectively




119899) for some 119897 gt 119899 Then

1003816100381610038161003816119877119898 (119860 119909 119910)1003816100381610038161003816

le 1198621003816100381610038161003816119909 minus 119910

1003816100381610038161003816

119898sum

|120574|=119898

(1

1003816100381610038161003816119868119910

119909

1003816100381610038161003816

int119868119910

119909

1003816100381610038161003816119863120574119860 (119911)

1003816100381610038161003816

119897119889119911)

1119897

(30)




10038171003817100381710038171198911003817100381710038171003817Λ120573

asymp sup119876

1

|119876|1+120573119899

int119876

1003816100381610038161003816119891 (119909) minus 119887119876 (119891)1003816100381610038161003816 119889119909

asymp sup119876

1

|119876|120573119899(1

|119876|int119876

1003816100381610038161003816119891 (119909) minus 119887119876 (119891)1003816100381610038161003816

119902119889119909)

1119902

(31)


Lemma 17 (see [13]) Let 1198761sub 119876

2 119892 isin Λ

120573(0 lt 120573 lt 1) Then

100381610038161003816100381610038161198921198761

minus 1198921198762

10038161003816100381610038161003816le 119862

100381610038161003816100381611987621003816100381610038161003816

120573119899 10038171003817100381710038171198921003817100381710038171003817Λ120573

(32)



119901(119908119901) to


conditions

(a) 1199041015840 lt 119901 and 119908(119909)1199041015840

isin 119860(1199011199041015840 119902119904

1015840)

(b) 119904 gt 119902 and 119908(119909)minus1199041015840

isin 119860(11990210158401199041015840 11990110158401199041015840)


such that 119908(119909)1199031015840

isin 119860(119901 119902)



119908 (2119861) ge 1198631119908 (119861) (33)





119894ge 2 and



(a) 1199041015840 lt 119901 and 119908(119909)1199041015840

isin 119860(1199011199041015840 119902119904

1015840)

(b) 119904 gt 119902 and 119908(119909)minus1199041015840

isin 119860(11990210158401199041015840 11990110158401199041015840)


1015840

isin 119860(119901 119902)


(intR119899

100381610038161003816100381610038161198791198601119860119896

Ω120572119891 (119909)119908 (119909)

10038161003816100381610038161003816

119902

119889119909)

1119902

le 119862

119896

prod

119894=1

( sum

|120574|=119898119894minus1

1003817100381710038171003817119863120574119860119894

1003817100381710038171003817lowast)

sdot (intR119899

1003816100381610038161003816119891 (119909)119908 (119909)1003816100381610038161003816

119901119889119909)

1119901

(34)


1

|119876|int119876

1003816100381610038161003816119887 minus 1198871198761003816100381610038161003816

119901119889119909 le 119862 119887

119901

lowast (35)



2

100381610038161003816100381610038161198871198761

minus 1198871198762

10038161003816100381610038161003816le 119862 log(

100381610038161003816100381611987621003816100381610038161003816

100381610038161003816100381611987611003816100381610038161003816

) 119887lowast (36)


1003816100381610038161003816 le 2119899(119895 + 1) 119887lowast (37)



119878119899minus1Ω(119909

1015840)119889120590(119909

1015840) = 0 If 119887 isin BMO(R119899) then [119887 119879

Ω]




isin 11986011990110158401199041015840

(c) 1 le 119901 lt infin and 1199081199041015840

isin 119860119901




Ω119891100381710038171003817100381710038171003817119871119901(119908)




2 More


119908 (120582119876) le 119862120582119899119901119908 (119876) (39)



119908 (119878)

119908 (119876)le 119862(

|119878|

|119876|)

120575

(40)



BMO (119908) = 119887 119887lowast119908

= sup119876

1

119908 (119876)int119876

1003816100381610038161003816119887 (119909) minus 1198981198761199081198871003816100381610038161003816 119908 (119909) 119889119909

119898119876119908119887 =

1

119908 (119876)int119876

119887 (119909)119908 (119909) 119889119909

(41)



Ω120572119891(119909) at first Set

119879Ω120572+120573

119891 (119909) = intR119899

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572minus120573

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

0 lt 120572 + 120573 lt 119899

(42)




10038161003816100381610038161003816119879119860

Ω120572119891 (119909)

10038161003816100381610038161003816le 119862( sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573

)119879Ω120572+120573

119891 (119909) (43)


119896= 2

119896119876 and set

119860119876119896

(119910) = 119860 (119910) minus sum

|120574|=119898minus1

1

120574119898119876119896

(119863120574119860)119910

120574 (44)

where 119898119876119896


|120574| = 119898 minus 1

119863120574119860119876119896

(119910) = 119863120574119860 (119910) minus 119898

119876119896

(119863120574119860) (45)


119877119898(119860 119909 119910) = 119877

119898(119860119876119896

119909 119910) (46)

Hence

10038161003816100381610038161003816119879119860

Ω120572119891 (119909)

10038161003816100381610038161003816

le

infin

sum

119896=minusinfin


119896119903

10038161003816100381610038161003816119877119898(119860119876119896

119909 119910)10038161003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119898minus1

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

flinfin

sum

119896=minusinfin

119879119896

(47)

By Lemma 15 we get

10038161003816100381610038161003816119877119898(119860119876119896

119909 119910)10038161003816100381610038161003816

le10038161003816100381610038161003816119877119898minus1

(119860119876119896

119909 119910)10038161003816100381610038161003816

+ 119862 sum

|120574|=119898minus1

10038161003816100381610038161003816119863120574119860119876119896

(119910)10038161003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119898minus1

le 1198621003816100381610038161003816119909 minus 119910

1003816100381610038161003816

119898minus1sum

|120574|=119898minus1

(1

1003816100381610038161003816119868119910

119909

1003816100381610038161003816

int119868119910

119909

10038161003816100381610038161003816119863120574119860119876119896

(119911)10038161003816100381610038161003816

119897

119889119911)

1119897

+ 1198621003816100381610038161003816119909 minus 119910

1003816100381610038161003816

119898minus1sum

|120574|=119898minus1

10038161003816100381610038161003816119863120574119860119876119896

(119910)10038161003816100381610038161003816

(48)



119896 By Lemmas 16


(1

1003816100381610038161003816119868119910

119909

1003816100381610038161003816

int119868119910

119909

10038161003816100381610038161003816119863120574119860119876119896

(119911)10038161003816100381610038161003816

119897

119889119911)

1119897

= (1

1003816100381610038161003816119868119910

119909

1003816100381610038161003816

int119868119910

119909

10038161003816100381610038161003816119863120574119860 (119911) minus 119898119876

119896

(119863120574119860)10038161003816100381610038161003816

119897

119889119911)

1119897

le (1

1003816100381610038161003816119868119910

119909

1003816100381610038161003816

int119868119910

119909

10038161003816100381610038161003816119863120574119860 (119911) minus 119898

119868119910

119909

(119863120574119860)10038161003816100381610038161003816

119897

119889119911)

1119897

+10038161003816100381610038161003816119898119868119910

119909

(119863120574119860) minus 119898

5119899119876119896

(119863120574119860)10038161003816100381610038161003816

+100381610038161003816100381610038161198985119899119876119896

(119863120574119860) minus 119898

119876119896

(119863120574119860)10038161003816100381610038161003816

le 11986210038161003816100381610038161198761198961003816100381610038161003816

120573119899 10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573

le 119862 (2119896119903)120573 1003817100381710038171003817119863

1205741198601003817100381710038171003817Λ120573

(49)


10038161003816100381610038161003816119863120574119860119876119896

(119910)10038161003816100381610038161003816=10038161003816100381610038161003816119863120574119860 (119910) minus 119898

119876119896

(119863120574119860)10038161003816100381610038161003816

le 11986210038161003816100381610038161198761198961003816100381610038161003816

120573119899 10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573

le 119862 (2119896119903)120573 1003817100381710038171003817119863

1205741198601003817100381710038171003817Λ120573

(50)

Thus

10038161003816100381610038161003816119877119898(119860119876119896

119909 119910)10038161003816100381610038161003816

le 1198621003816100381610038161003816119909 minus 119910

1003816100381610038161003816

119898minus1(2119896119903)120573

sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573

(51)

Therefore

119879119896le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573


119896119903

(2119896119903)120573

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572

1003816100381610038161003816Ω (119909 minus 119910)10038161003816100381610038161003816100381610038161003816119891 (119910)

1003816100381610038161003816 119889119910

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573


119896119903

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572minus120573

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

(52)

It follows that

10038161003816100381610038161003816119879119860

Ω120572119891 (119909)

10038161003816100381610038161003816le

infin

sum

119896=minusinfin

(119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573


119896119903

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572minus120573

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910)

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573

sdot

infin

sum

119896=minusinfin


119896119903

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572minus120573

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

= 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573

intR119899

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572minus120573

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

= 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573

119879Ω120572+120573

119891 (119909)

(53)






1+ 119891

2with

1198911= 119891120594


(1

119908119902 (119861)120581119902119901

int119861

10038161003816100381610038161003816119879Ω120572+120573

119891 (119909)10038161003816100381610038161003816

119902

119908119902(119909) 119889119909)

1119902

le1

119908119902 (119861)120581119901(int119861

10038161003816100381610038161003816119879Ω120572+120573

1198911 (119909)

10038161003816100381610038161003816

119902

119908119902(119909) 119889119909)

1119902

+1

119908119902 (119861)120581119901(int119861

10038161003816100381610038161003816119879Ω120572+120573

1198912 (119909)

10038161003816100381610038161003816

119902

119908119902(119909) 119889119909)

1119902

fl 1198691+ 1198692

(54)


2


1198691le

1

119908119902 (119861)120581119901

10038171003817100381710038171003817119879Ω120572+120573

1198911

10038171003817100381710038171003817119871119902(119908119902)

le119862

119908119902 (119861)120581119901

100381710038171003817100381711989111003817100381710038171003817119871119901(119908119901)

=119862

119908119902 (119861)120581119901(int2119861

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119901119908 (119909)

119901119889119909)

1119901

le 11986210038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

119908119902(2119861)

120581119901

119908119902 (119861)120581119901

le 11986210038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

(55)



10038161003816100381610038161003816119879Ω120572+120573

1198912 (119909)

10038161003816100381610038161003816=

infin

sum

119895=1

int2119895+11198612119895119861

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572minus120573

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

le 119862

infin

sum

119895=1

(int2119895+1119861

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

119904119889119910)

1119904

sdot (int2119895+11198612119895119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816

1199041015840

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

(119899minus120572minus120573)1199041015840119889119910)

11199041015840

fl 119862infin

sum

119895=1

(11986811198951198682119895)

(56)



1198681119895= (int

2119895+2119861

|Ω (119911)|119904119889119911)

1119904

= (int

2119895+2

119903119861

0

int119878119899minus1

10038161003816100381610038161003816Ω (119911

1015840)10038161003816100381610038161003816

119904

1198891199111015840119903119899minus1119889119903)

1119904

= 119862 Ω119871119904(119878119899minus1)

100381610038161003816100381610038162119895+211986110038161003816100381610038161003816

1119904

(57)


1198682119895le

119862

10038161003816100381610038162119895+1119861

1003816100381610038161003816

1minus(120572+120573)119899(int2119895+1119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816

1199041015840

119889119910)

11199041015840

(58)


isin 119860(1199011199041015840 119902119904

1015840) we get

(int2119895+1119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816

1199041015840

119889119910)

11199041015840

le 119862(int2119895+1119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119901119908 (119910)

119901119889119910)

1119901

sdot (int2119895+1119861

119908 (119910)minus1199011199041015840

(119901minus1199041015840

)119889119910)

(119901minus1199041015840

)1199011199041015840

le 11986210038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) 119908

119902(2119895+1119861)120581119901

sdot (int2119895+1119861

119908 (119910)minus1199011199041015840

(119901minus1199041015840

)119889119910)

(119901minus1199041015840

)1199011199041015840

le 11986210038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) 119908

119902(2119895+1119861)120581119901

sdot

100381610038161003816100381610038162119895+111986110038161003816100381610038161003816

(119901119902minus1199041015840

119902+1199041015840

119901)1199011199021199041015840

119908119902 (2119895+1119861)1119902

(59)

Thus

10038161003816100381610038161003816119879Ω120572+120573

1198912 (119909)

10038161003816100381610038161003816le 119862

infin

sum

119895=1

(11986811198951198682119895)

le 119862

infin

sum

119895=1

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

1

119908119902 (2119895+1119861)1119902minus120581119901

(60)

So we get

1198692le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

infin

sum

119895=1

119908119902(119861)

1119902minus120581119901

119908119902 (2119895+1119861)1119902minus120581119901

(61)


1198692le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) (62)



Ω120572is bounded




119879119860

Ω120572119891 (119909)

= intR119899

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572+119898minus1

1003816100381610038161003816119877119898 (119860 119909 119910)10038161003816100381610038161003816100381610038161003816119891 (119910)

1003816100381610038161003816 119889119910

0 le 120572 lt 119899

(63)


119860

Ω120572 On the


119879119860

Ω120572119891 (119909)

ge int|119909minus119910|lt119903

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572+119898minus1

1003816100381610038161003816119877119898 (119860 119909 119910)10038161003816100381610038161003816100381610038161003816119891 (119910)

1003816100381610038161003816 119889119910

ge1

119903119899minus120572+119898minus1


1003816100381610038161003816Ω (119909 minus 119910)10038161003816100381610038161003816100381610038161003816119877119898 (119860 119909 119910)

10038161003816100381610038161003816100381610038161003816119891 (119910)

1003816100381610038161003816 119889119910

(64)


119879119860

Ω120572119891 (119909) ge 119872

119860

Ω120572119891 (119909) (65)




119879Ω120573119891 (119909) = int

R119899

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120573

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910 (66)







119860119876(119910) = 119860 (119910) minus sum

|120574|=119898minus1

1

120574119898119876(119863120574119860)119910

120574 (67)


119876+ 119891120594

(119876)119888 = 119891

1+ 119891

2

Then we have

1

119908119902 (119876)120581119901(int119876

10038161003816100381610038161003816119879119860

Ω120572119891 (119910)

10038161003816100381610038161003816

119902

119908 (119910)119902119889119910)

1119902

le1

119908119902 (119876)120581119901(int119876

10038161003816100381610038161003816119879119860

Ω1205721198911(119910)10038161003816100381610038161003816

119902

119908 (119910)119902119889119910)

1119902

+1

119908119902 (119876)120581119901(int119876

10038161003816100381610038161003816119879119860

Ω1205721198912(119910)10038161003816100381610038161003816

119902

119908 (119910)119902119889119910)

1119902

fl 119868 + 119868119868

(68)


119868 le119862

119908119902 (119876)120581119901

sdot sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast (int

119876

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119901119908 (119910)

119901119889119910)

1119901

= 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)(

119908119902(119876)

119908119902 (119876))

120581119901

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

(69)


Ω1205721198912(119910) contained in 119868119868 By


10038161003816100381610038161003816119879119860

Ω1205721198912(119910)10038161003816100381610038161003816

le int(119876)119888

10038161003816100381610038161003816119877119898(119860119876 119910 119911)

100381610038161003816100381610038161003816100381610038161003816119910 minus 119911

1003816100381610038161003816

119898minus1

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

le 119862int(119876)119888

sum

|120574|=119898minus1

(1

10038161003816100381610038161003816119868119911119910

10038161003816100381610038161003816

int119868119911

119910

10038161003816100381610038161003816119863120574119860119876(119905)10038161003816100381610038161003816

119897

119889119905)

1119897

sdot1003816100381610038161003816Ω (119910 minus 119911)

1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572119889119911

+ 119862int(119876)119888

sum

|120574|=119898minus1

10038161003816100381610038161003816119863120574119860 (119911) minus 119898

119876(119863120574119860)10038161003816100381610038161003816

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

sdot

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572119889119911 fl 119868119868

1+ 119868119868

2

(70)

We estimate 1198681198681and 119868119868



isin 119860(1199011199041015840 119902119904

1015840) we get

1198681198681le 119862 sum

|120574|=119898minus1

(1

10038161003816100381610038161003816119868119911119910

10038161003816100381610038161003816

int119868119911

119910

10038161003816100381610038161003816119863120574119860 (119905) minus 119898

119876(119863120574119860)10038161003816100381610038161003816

119897

119889119905)

1119897

sdot

infin

sum

119895=1

int2119895+11198762119895119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572119889119911

le 119862 sum

|120574|=119898minus1

[

[

(1

10038161003816100381610038161003816119868119911119910

10038161003816100381610038161003816

sdot int119868119911

119910

100381610038161003816100381610038161003816119863120574119860 (119905) minus 119898119868119911

119910

(119863120574119860)100381610038161003816100381610038161003816

119897

119889119905)

1119897

+100381610038161003816100381610038161003816119898119868119911

119910

(119863120574119860)

minus 1198985119899119876(119863120574119860)100381610038161003816100381610038161003816+100381610038161003816100381610038161198985119899119876(119863120574119860) minus 119898

119876(119863120574119860)10038161003816100381610038161003816]

]

sdot

infin

sum

119895=1

(int2119895+1119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

119904119889119911)

1119904

sdot (int2119895+11198762119895119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

(119899minus120572)1199041015840119889119911)

11199041015840

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast Ω119871

119904(119878119899minus1)

sdot

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus120572119899(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840


le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus120572119899(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

119901

sdot 119908 (119911)119901119889119911)

1119901

sdot (int2119895+1119876

119908 (119911)minus1199011199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

sdot

infin

sum

119895=1

1

119908119902 (2119895+1119876)1119902minus120581119901

(71)


1198681198682le 119862

infin

sum

119895=1

int2119895+11198762119895119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

sdot1003816100381610038161003816Ω (119910 minus 119911)

1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572119889119911

+ 119862

infin

sum

119895=1

int2119895+11198762119895119876

sum

|120574|=119898minus1

100381610038161003816100381610038161198982119895+1119876(119863120574119860) minus 119898

119876(119863120574119860)10038161003816100381610038161003816

sdot1003816100381610038161003816Ω (119910 minus 119911)

1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572119889119911 fl 119868119868

21+ 119868119868

22

(72)


11986811986821le 119862

infin

sum

119895=1

1

100381610038161003816100381621198951198761003816100381610038161003816

1minus

120572

119899

int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863

120574119860)10038161003816100381610038161003816100381610038161003816Ω (119910 minus 119911)

10038161003816100381610038161003816100381610038161003816119891 (119911)

1003816100381610038161003816 119889119911

le 119862

infin

sum

119895=1

1

100381610038161003816100381621198951198761003816100381610038161003816

1minus

120572

119899

(int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

1199041015840

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840

(int2119895+1119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

119904119889119911)

1119904

le 119862 Ω119871119904(119878119899minus1)

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus

120572

119899

(int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

1199041015840

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840

le 119862 Ω119871119904(119878119899minus1)

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus

120572

119899

(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

119901119908 (119911)

119901119889119911)

1119901

sdot (int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

1199011199041015840

(119901minus1199041015840

)119908 (119911)

minus1199011199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

(73)


(int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

1199011199041015840

(119901minus1199041015840

)119908 (119911)

minus1199011199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

le 119862(int2119895+1119876

sum

|120574|=119898minus1

10038161003816100381610038161003816119863120574119860 (119911) minus 119898

2119895+1119876119908minus1199011199041015840(119901minus1199041015840)(119863120574119860)10038161003816100381610038161003816

1199011199041015840

(119901minus1199041015840

)

119908 (119911)minus1199011199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

+ sum

|120574|=119898minus1


2119895+1119876(119863120574119860)10038161003816100381610038161003816119908minus1199011199041015840

(119901minus1199041015840

)(2119895+1119876)(119901minus1199041015840

)1199011199041015840

fl 119868119868119868 + 119868119881

(74)


isin 119860(1199011199041015840 119902119904


119908minus1199011199041015840

(119901minus1199041015840

)isin 119860

1199051015840 sub 119860



(119901minus1199041015840


of BMO(R119899) and 1199081199041015840

isin 119860(1199011199041015840 119902119904

1015840) we have


119868119868119868 le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast 119908

minus1199011199041015840

(119901minus1199041015840

)(2119895+1119876)(119901minus1199041015840

)1199011199041015840

= 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

100381610038161003816100381610038162119895+111987610038161003816100381610038161003816

(119901minus1199041015840

)1199011199041015840

+1119902

119908119902 (2119895+1119876)1119902

(75)



119905 isin 2119895+1119876 sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816 gt 119904

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le 1198621

100381610038161003816100381610038162119895+111987610038161003816100381610038161003816119890minus1198622119904(sum|120574|=119898minus1

119863120574

119860lowast)

(76)



119908(

119905 isin 2119895+1119876 sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

gt 119904

) le 119862119908(2119895+1119876) 119890

minus1198622119904120575(sum

|120574|=119898minus1119863120574

119860lowast)

(77)


sum

|120574|=119898minus1


2119895+1119876(119863120574119860)10038161003816100381610038161003816

le1

119908minus1199011199041015840(119901minus119904

1015840) (2119895+1119876)

int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119905)

minus 1198982119895+1119876(119863120574119860)1003816100381610038161003816 119908minus1199011199041015840

(119901minus1199041015840

)(119905) 119889119905

=119862

119908minus1199011199041015840(119901minus119904

1015840) (2119895+1119876)

int

infin

0

119908minus1199011199041015840

(119901minus1199041015840

)(

119905

isin 2119895+1119876 sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

gt 119904

)119889119904 le119862

119908minus1199011199041015840(119901minus119904

1015840) (2119895+1119876)

sdot int

infin

0

119908minus1199011199041015840

(119901minus1199041015840

)(2119895+1119876) 119890

minus1198622119904120575(sum

|120574|=119898minus1119863120574

119860lowast)119889119904

= 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

(78)

As a result

119868119881 le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast 119908

minus1199011199041015840

(119901minus1199041015840

)(2119895+1119876)(119901minus1199041015840

)1199011199041015840

= 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

100381610038161003816100381610038162119895+111987610038161003816100381610038161003816

(119901minus1199041015840

)1199011199041015840

+1119902

119908119902 (2119895+1119876)1119902

(79)

Thus

11986811986821le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

sdot

infin

sum

119895=1

1

119908119902 (2119895+1119876)1119902minus120581119901

(80)


1199081199041015840

isin 119860(1199011199041015840 119902119904

1015840) we get

11986811986822le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

sdot

infin

sum

119895=1

119895 int2119895+11198762119895119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572119889119911

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast Ω119871

119904(119878119899minus1)

infin

sum

119895=1

119895

sdot

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus120572119899(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

infin

sum

119895=1

119895

sdot

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus120572119899(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

119901119908 (119911)

119901119889119911)

1119901

sdot (int2119895+1119876

119908 (119911)minus1199011199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

sdot

infin

sum

119895=1

119895

119908119902 (2119895+1119876)1119902minus120581119901

(81)

Hence

10038161003816100381610038161003816119879119860

Ω1205721198912(119910)10038161003816100381610038161003816le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

sdot

infin

sum

119895=1

119895

119908119902 (2119895+1119876)1119902minus120581119901

(82)

Therefore

119868119868 le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

sdot

infin

sum

119895=1

119895119908119902(119876)

1119902minus120581119901

119908119902 (2119895+1119876)1119902minus120581119901


le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

infin

sum

119895=1

119895

(119863119895+1

1)1119902minus120581119901

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

(83)


1

119908119902 (119876)120581119901(int119876

10038161003816100381610038161003816119879119860

Ω120572119891 (119910)

10038161003816100381610038161003816

119902

119908 (119910)119902119889119910)

1119902

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

(84)




119876+ 119891120594

(119876)119888 fl 119891

1+ 119891

2 Then we have

1

119908 (119876)120581119901(int119876

1003816100381610038161003816[119860 119879Ω] 119891 (119910)1003816100381610038161003816

119901119908 (119910) 119889119910)

1119901

le1

119908 (119876)120581119901(int119876

1003816100381610038161003816[119860 119879Ω] 1198911 (119910)1003816100381610038161003816

119901119908 (119910) 119889119910)

1119901

+1

119908 (119876)120581119901(int119876

1003816100381610038161003816[119860 119879Ω] 1198912 (119910)1003816100381610038161003816

119901119908 (119910) 119889119910)

1119901

fl 119868 + 119868119868

(85)


119868 le1

119908 (119876)120581119901

1003817100381710038171003817[119860 119879Ω] 11989111003817100381710038171003817119871119901(119908)

le119862

119908 (119876)120581119901119860lowast

100381710038171003817100381711989111003817100381710038171003817119871119901(119908)

= 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

119908(119876)120581119901

119908 (119876)120581119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(86)


1003816100381610038161003816[119860 119879Ω] 1198912 (119910)1003816100381610038161003816 le

infin

sum

119895=1

int2119895+11198762119895119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899

1003816100381610038161003816119860 (119910) minus 119860 (119911)1003816100381610038161003816

sdot1003816100381610038161003816119891 (119911)

1003816100381610038161003816 119889119911

le 119862

infin

sum

119895=1

1

100381610038161003816100381621198951198761003816100381610038161003816

(int2119895+1119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

119904119889119911)

1119904

sdot (int2119895+1119876

1003816100381610038161003816119860 (119910) minus 119860 (119911)1003816100381610038161003816

1199041015840

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840

le 119862 Ω119871119904

sdot

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

sdot (int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840

+ 119862 Ω119871119904

sdot

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

(int2119895+1119876


10038161003816100381610038161003816

1199041015840

sdot1003816100381610038161003816119891 (119911)

1003816100381610038161003816

1199041015840

119889119911)

11199041015840

fl 1198681198681(119910) + 119868119868

2

(87)




1

119908 (119876)120581119901(int119876

1198681198681(119910)

119901119908 (119910) 119889119910)

1119901

= 119862Ω119871119904

119908 (119876)120581119901

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

(int119876

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

119901

(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

1199011199041015840

119908 (119910) 119889119910)

1119901

le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

119908 (119876)

120581

119901

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

119908 (2119895+1119876)120581119901

119908minus1199041015840

(119901minus1199041015840

)(2119895+1119876)(119901minus1199041015840

)1199011199041015840

sdot (int119876

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

119908 (119876)120581119901

infin

sum

119895=1

1

119908 (2119895+1119876)(1minus120581)119901

(int119876

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

(88)



(int119876

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

le (int119876

1003816100381610038161003816119860 (119910) minus 119898119876119908 (119860)1003816100381610038161003816

119901119908 (119910) 119889119910)

1119901

+10038161003816100381610038161003816119898119876119908 (119860) minus 1198982119895+1119876119908minus119904

1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816119908 (119876)

1119901

fl 119868119868119868 + 119868119881

(89)



Lemma 25 that

119868119868119868 le 119860lowast 119908 (119876)1119901 (90)


10038161003816100381610038161003816119898119876119908 (119860) minus 1198982119895+1119876119908minus119904

1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

le1003816100381610038161003816119898119876119908 (119860) minus 119898119876 (119860)

1003816100381610038161003816 +1003816100381610038161003816119898119876 (119860) minus 1198982119895+1119876 (119860)

1003816100381610038161003816

+100381610038161003816100381610038161198982119895+1119876 (119860) minus 1198982119895+1119876119908minus119904

1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816le

1

119908 (119876)

sdot int119876

1003816100381610038161003816119860 (119905) minus 119898119876 (119860)1003816100381610038161003816 119908 (119905) 119889119905 + 2

119899(119895 + 1) 119860lowast

+1

119908minus1199041015840(119901minus119904

1015840) (2119895+1119876)

sdot int2119895+1119876

1003816100381610038161003816119860 (119905) minus 1198982119895+1119876 (119860)1003816100381610038161003816 119908minus1199041015840

(119901minus1199041015840

)(119905) 119889119905

le 119862 (119895 + 1) 119860lowast

(91)

Hence

119868119881 le 119862 (119895 + 1) 119860lowast 119908 (119876)1119901 (92)

As a result

1

119908 (119876)120581119901(int119876

1198681198681(119910)

119901119908 (119910) 119889119910)

1119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

infin

sum

119895=1

(119895 + 1)119908 (119876)

(1minus120581)119901

119908 (2119895+1119876)(1minus120581)119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(93)


1199011199041015840 we get

1198681198682le 119862 Ω119871119904

sdot

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

119901119908 (119911) 119889119911)

1119901

sdot (int2119895+1119876


10038161003816100381610038161003816

1199011199041015840

(119901minus1199041015840

)

sdot 119908 (119911)minus1199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

le 119862 Ω119871119904 119860lowast

sdot10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

119908 (2119895+1119876)120581119901

sdot 119908minus1199041015840

(119901minus1199041015840

)(2119895+1119876)(119901minus1199041015840

)1199011199041015840

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

sdot

infin

sum

119895=1

1

119908 (2119895+1119876)(1minus120581)119901

(94)

Therefore

1

119908 (119876)120581119901(int119876

119868119868119901

2119908 (119910) 119889119910)

1119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

infin

sum

119895=1

119908 (119876)(1minus120581)119901

119908 (2119895+1119876)(1minus120581)119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(95)




119860119876(119910) = 119860 (119910) minus 119898

119876(nabla119860) 119910 (96)


(119876)119888 fl

1198911+ 119891

2 Then we get

1

119908 (119876)120581119901(int119876

100381610038161003816100381610038161003816119860

Ω119891 (119910)

100381610038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

le1

119908 (119876)120581119901(int119876

100381610038161003816100381610038161003816119860

Ω1198911(119910)100381610038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

+1

119908 (119876)120581119901(int119876

100381610038161003816100381610038161003816119860

Ω1198912(119910)100381610038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

fl 119868 + 119868119868

(97)



119868 le1

119908 (119876)120581119901

100381710038171003817100381710038171003817119860

Ω1198911

100381710038171003817100381710038171003817119871119901(119908)

le119862


100381710038171003817100381711989111003817100381710038171003817119871119901(119908)

le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

119908(119876)120581119901

119908 (119876)120581119901

le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(98)


119901 Ω isin 119871




Competing Interests


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119878119899minus1120579Ω(120579)119889120579 = 0 1 lt 119901 lt


100381710038171003817100381710038171003817119860

Ω119891100381710038171003817100381710038171003817119871119901120581(119908)

le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908) (27)

100381710038171003817100381710038171003817119860

Ω119891100381710038171003817100381710038171003817119871119901120581(119908)

le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908) (28)


Ω when



Remark 14 Define

1198791198601119860119896

Ω120572119891 (119909) = int

R119899

119896

prod

119894=1

119877119898119894

(119860119894 119909 119910)

sdotΩ (119909 minus 119910)

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572+119873119891 (119910) 119889119910

1198721198601119860119896

Ω120572119891 (119909) = sup

119903gt0

1


1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

sdot

119896

prod

119894=1

10038161003816100381610038161003816119877119898119894

(119860119894 119909 119910)

10038161003816100381610038161003816

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

(29)

where 119877119898119894

(119860119894 119909 119910) = 119860

119894(119909) minus sum

|120574|lt119898119894

(1120574)119863120574119860119894(119910)(119909 minus

119910)120574 119894 = 1 119896 119873 = sum

119896

119894=1(119898119894minus 1) When 0 lt 120572 lt 119899


Ω120572and 119872119860

1119860119896



119894minus1119863120574119860119894Λ120573


119894=1(sum|120574|=119898


hold respectively




119899) for some 119897 gt 119899 Then

1003816100381610038161003816119877119898 (119860 119909 119910)1003816100381610038161003816

le 1198621003816100381610038161003816119909 minus 119910

1003816100381610038161003816

119898sum

|120574|=119898

(1

1003816100381610038161003816119868119910

119909

1003816100381610038161003816

int119868119910

119909

1003816100381610038161003816119863120574119860 (119911)

1003816100381610038161003816

119897119889119911)

1119897

(30)




10038171003817100381710038171198911003817100381710038171003817Λ120573

asymp sup119876

1

|119876|1+120573119899

int119876

1003816100381610038161003816119891 (119909) minus 119887119876 (119891)1003816100381610038161003816 119889119909

asymp sup119876

1

|119876|120573119899(1

|119876|int119876

1003816100381610038161003816119891 (119909) minus 119887119876 (119891)1003816100381610038161003816

119902119889119909)

1119902

(31)


Lemma 17 (see [13]) Let 1198761sub 119876

2 119892 isin Λ

120573(0 lt 120573 lt 1) Then

100381610038161003816100381610038161198921198761

minus 1198921198762

10038161003816100381610038161003816le 119862

100381610038161003816100381611987621003816100381610038161003816

120573119899 10038171003817100381710038171198921003817100381710038171003817Λ120573

(32)



119901(119908119901) to


conditions

(a) 1199041015840 lt 119901 and 119908(119909)1199041015840

isin 119860(1199011199041015840 119902119904

1015840)

(b) 119904 gt 119902 and 119908(119909)minus1199041015840

isin 119860(11990210158401199041015840 11990110158401199041015840)


such that 119908(119909)1199031015840

isin 119860(119901 119902)



119908 (2119861) ge 1198631119908 (119861) (33)





119894ge 2 and



(a) 1199041015840 lt 119901 and 119908(119909)1199041015840

isin 119860(1199011199041015840 119902119904

1015840)

(b) 119904 gt 119902 and 119908(119909)minus1199041015840

isin 119860(11990210158401199041015840 11990110158401199041015840)


1015840

isin 119860(119901 119902)


(intR119899

100381610038161003816100381610038161198791198601119860119896

Ω120572119891 (119909)119908 (119909)

10038161003816100381610038161003816

119902

119889119909)

1119902

le 119862

119896

prod

119894=1

( sum

|120574|=119898119894minus1

1003817100381710038171003817119863120574119860119894

1003817100381710038171003817lowast)

sdot (intR119899

1003816100381610038161003816119891 (119909)119908 (119909)1003816100381610038161003816

119901119889119909)

1119901

(34)


1

|119876|int119876

1003816100381610038161003816119887 minus 1198871198761003816100381610038161003816

119901119889119909 le 119862 119887

119901

lowast (35)



2

100381610038161003816100381610038161198871198761

minus 1198871198762

10038161003816100381610038161003816le 119862 log(

100381610038161003816100381611987621003816100381610038161003816

100381610038161003816100381611987611003816100381610038161003816

) 119887lowast (36)


1003816100381610038161003816 le 2119899(119895 + 1) 119887lowast (37)



119878119899minus1Ω(119909

1015840)119889120590(119909

1015840) = 0 If 119887 isin BMO(R119899) then [119887 119879

Ω]




isin 11986011990110158401199041015840

(c) 1 le 119901 lt infin and 1199081199041015840

isin 119860119901




Ω119891100381710038171003817100381710038171003817119871119901(119908)




2 More


119908 (120582119876) le 119862120582119899119901119908 (119876) (39)



119908 (119878)

119908 (119876)le 119862(

|119878|

|119876|)

120575

(40)



BMO (119908) = 119887 119887lowast119908

= sup119876

1

119908 (119876)int119876

1003816100381610038161003816119887 (119909) minus 1198981198761199081198871003816100381610038161003816 119908 (119909) 119889119909

119898119876119908119887 =

1

119908 (119876)int119876

119887 (119909)119908 (119909) 119889119909

(41)



Ω120572119891(119909) at first Set

119879Ω120572+120573

119891 (119909) = intR119899

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572minus120573

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

0 lt 120572 + 120573 lt 119899

(42)




10038161003816100381610038161003816119879119860

Ω120572119891 (119909)

10038161003816100381610038161003816le 119862( sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573

)119879Ω120572+120573

119891 (119909) (43)


119896= 2

119896119876 and set

119860119876119896

(119910) = 119860 (119910) minus sum

|120574|=119898minus1

1

120574119898119876119896

(119863120574119860)119910

120574 (44)

where 119898119876119896


|120574| = 119898 minus 1

119863120574119860119876119896

(119910) = 119863120574119860 (119910) minus 119898

119876119896

(119863120574119860) (45)


119877119898(119860 119909 119910) = 119877

119898(119860119876119896

119909 119910) (46)

Hence

10038161003816100381610038161003816119879119860

Ω120572119891 (119909)

10038161003816100381610038161003816

le

infin

sum

119896=minusinfin


119896119903

10038161003816100381610038161003816119877119898(119860119876119896

119909 119910)10038161003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119898minus1

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

flinfin

sum

119896=minusinfin

119879119896

(47)

By Lemma 15 we get

10038161003816100381610038161003816119877119898(119860119876119896

119909 119910)10038161003816100381610038161003816

le10038161003816100381610038161003816119877119898minus1

(119860119876119896

119909 119910)10038161003816100381610038161003816

+ 119862 sum

|120574|=119898minus1

10038161003816100381610038161003816119863120574119860119876119896

(119910)10038161003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119898minus1

le 1198621003816100381610038161003816119909 minus 119910

1003816100381610038161003816

119898minus1sum

|120574|=119898minus1

(1

1003816100381610038161003816119868119910

119909

1003816100381610038161003816

int119868119910

119909

10038161003816100381610038161003816119863120574119860119876119896

(119911)10038161003816100381610038161003816

119897

119889119911)

1119897

+ 1198621003816100381610038161003816119909 minus 119910

1003816100381610038161003816

119898minus1sum

|120574|=119898minus1

10038161003816100381610038161003816119863120574119860119876119896

(119910)10038161003816100381610038161003816

(48)



119896 By Lemmas 16


(1

1003816100381610038161003816119868119910

119909

1003816100381610038161003816

int119868119910

119909

10038161003816100381610038161003816119863120574119860119876119896

(119911)10038161003816100381610038161003816

119897

119889119911)

1119897

= (1

1003816100381610038161003816119868119910

119909

1003816100381610038161003816

int119868119910

119909

10038161003816100381610038161003816119863120574119860 (119911) minus 119898119876

119896

(119863120574119860)10038161003816100381610038161003816

119897

119889119911)

1119897

le (1

1003816100381610038161003816119868119910

119909

1003816100381610038161003816

int119868119910

119909

10038161003816100381610038161003816119863120574119860 (119911) minus 119898

119868119910

119909

(119863120574119860)10038161003816100381610038161003816

119897

119889119911)

1119897

+10038161003816100381610038161003816119898119868119910

119909

(119863120574119860) minus 119898

5119899119876119896

(119863120574119860)10038161003816100381610038161003816

+100381610038161003816100381610038161198985119899119876119896

(119863120574119860) minus 119898

119876119896

(119863120574119860)10038161003816100381610038161003816

le 11986210038161003816100381610038161198761198961003816100381610038161003816

120573119899 10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573

le 119862 (2119896119903)120573 1003817100381710038171003817119863

1205741198601003817100381710038171003817Λ120573

(49)


10038161003816100381610038161003816119863120574119860119876119896

(119910)10038161003816100381610038161003816=10038161003816100381610038161003816119863120574119860 (119910) minus 119898

119876119896

(119863120574119860)10038161003816100381610038161003816

le 11986210038161003816100381610038161198761198961003816100381610038161003816

120573119899 10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573

le 119862 (2119896119903)120573 1003817100381710038171003817119863

1205741198601003817100381710038171003817Λ120573

(50)

Thus

10038161003816100381610038161003816119877119898(119860119876119896

119909 119910)10038161003816100381610038161003816

le 1198621003816100381610038161003816119909 minus 119910

1003816100381610038161003816

119898minus1(2119896119903)120573

sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573

(51)

Therefore

119879119896le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573


119896119903

(2119896119903)120573

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572

1003816100381610038161003816Ω (119909 minus 119910)10038161003816100381610038161003816100381610038161003816119891 (119910)

1003816100381610038161003816 119889119910

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573


119896119903

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572minus120573

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

(52)

It follows that

10038161003816100381610038161003816119879119860

Ω120572119891 (119909)

10038161003816100381610038161003816le

infin

sum

119896=minusinfin

(119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573


119896119903

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572minus120573

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910)

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573

sdot

infin

sum

119896=minusinfin


119896119903

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572minus120573

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

= 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573

intR119899

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572minus120573

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

= 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573

119879Ω120572+120573

119891 (119909)

(53)






1+ 119891

2with

1198911= 119891120594


(1

119908119902 (119861)120581119902119901

int119861

10038161003816100381610038161003816119879Ω120572+120573

119891 (119909)10038161003816100381610038161003816

119902

119908119902(119909) 119889119909)

1119902

le1

119908119902 (119861)120581119901(int119861

10038161003816100381610038161003816119879Ω120572+120573

1198911 (119909)

10038161003816100381610038161003816

119902

119908119902(119909) 119889119909)

1119902

+1

119908119902 (119861)120581119901(int119861

10038161003816100381610038161003816119879Ω120572+120573

1198912 (119909)

10038161003816100381610038161003816

119902

119908119902(119909) 119889119909)

1119902

fl 1198691+ 1198692

(54)


2


1198691le

1

119908119902 (119861)120581119901

10038171003817100381710038171003817119879Ω120572+120573

1198911

10038171003817100381710038171003817119871119902(119908119902)

le119862

119908119902 (119861)120581119901

100381710038171003817100381711989111003817100381710038171003817119871119901(119908119901)

=119862

119908119902 (119861)120581119901(int2119861

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119901119908 (119909)

119901119889119909)

1119901

le 11986210038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

119908119902(2119861)

120581119901

119908119902 (119861)120581119901

le 11986210038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

(55)



10038161003816100381610038161003816119879Ω120572+120573

1198912 (119909)

10038161003816100381610038161003816=

infin

sum

119895=1

int2119895+11198612119895119861

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572minus120573

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

le 119862

infin

sum

119895=1

(int2119895+1119861

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

119904119889119910)

1119904

sdot (int2119895+11198612119895119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816

1199041015840

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

(119899minus120572minus120573)1199041015840119889119910)

11199041015840

fl 119862infin

sum

119895=1

(11986811198951198682119895)

(56)



1198681119895= (int

2119895+2119861

|Ω (119911)|119904119889119911)

1119904

= (int

2119895+2

119903119861

0

int119878119899minus1

10038161003816100381610038161003816Ω (119911

1015840)10038161003816100381610038161003816

119904

1198891199111015840119903119899minus1119889119903)

1119904

= 119862 Ω119871119904(119878119899minus1)

100381610038161003816100381610038162119895+211986110038161003816100381610038161003816

1119904

(57)


1198682119895le

119862

10038161003816100381610038162119895+1119861

1003816100381610038161003816

1minus(120572+120573)119899(int2119895+1119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816

1199041015840

119889119910)

11199041015840

(58)


isin 119860(1199011199041015840 119902119904

1015840) we get

(int2119895+1119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816

1199041015840

119889119910)

11199041015840

le 119862(int2119895+1119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119901119908 (119910)

119901119889119910)

1119901

sdot (int2119895+1119861

119908 (119910)minus1199011199041015840

(119901minus1199041015840

)119889119910)

(119901minus1199041015840

)1199011199041015840

le 11986210038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) 119908

119902(2119895+1119861)120581119901

sdot (int2119895+1119861

119908 (119910)minus1199011199041015840

(119901minus1199041015840

)119889119910)

(119901minus1199041015840

)1199011199041015840

le 11986210038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) 119908

119902(2119895+1119861)120581119901

sdot

100381610038161003816100381610038162119895+111986110038161003816100381610038161003816

(119901119902minus1199041015840

119902+1199041015840

119901)1199011199021199041015840

119908119902 (2119895+1119861)1119902

(59)

Thus

10038161003816100381610038161003816119879Ω120572+120573

1198912 (119909)

10038161003816100381610038161003816le 119862

infin

sum

119895=1

(11986811198951198682119895)

le 119862

infin

sum

119895=1

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

1

119908119902 (2119895+1119861)1119902minus120581119901

(60)

So we get

1198692le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

infin

sum

119895=1

119908119902(119861)

1119902minus120581119901

119908119902 (2119895+1119861)1119902minus120581119901

(61)


1198692le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) (62)



Ω120572is bounded




119879119860

Ω120572119891 (119909)

= intR119899

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572+119898minus1

1003816100381610038161003816119877119898 (119860 119909 119910)10038161003816100381610038161003816100381610038161003816119891 (119910)

1003816100381610038161003816 119889119910

0 le 120572 lt 119899

(63)


119860

Ω120572 On the


119879119860

Ω120572119891 (119909)

ge int|119909minus119910|lt119903

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572+119898minus1

1003816100381610038161003816119877119898 (119860 119909 119910)10038161003816100381610038161003816100381610038161003816119891 (119910)

1003816100381610038161003816 119889119910

ge1

119903119899minus120572+119898minus1


1003816100381610038161003816Ω (119909 minus 119910)10038161003816100381610038161003816100381610038161003816119877119898 (119860 119909 119910)

10038161003816100381610038161003816100381610038161003816119891 (119910)

1003816100381610038161003816 119889119910

(64)


119879119860

Ω120572119891 (119909) ge 119872

119860

Ω120572119891 (119909) (65)




119879Ω120573119891 (119909) = int

R119899

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120573

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910 (66)







119860119876(119910) = 119860 (119910) minus sum

|120574|=119898minus1

1

120574119898119876(119863120574119860)119910

120574 (67)


119876+ 119891120594

(119876)119888 = 119891

1+ 119891

2

Then we have

1

119908119902 (119876)120581119901(int119876

10038161003816100381610038161003816119879119860

Ω120572119891 (119910)

10038161003816100381610038161003816

119902

119908 (119910)119902119889119910)

1119902

le1

119908119902 (119876)120581119901(int119876

10038161003816100381610038161003816119879119860

Ω1205721198911(119910)10038161003816100381610038161003816

119902

119908 (119910)119902119889119910)

1119902

+1

119908119902 (119876)120581119901(int119876

10038161003816100381610038161003816119879119860

Ω1205721198912(119910)10038161003816100381610038161003816

119902

119908 (119910)119902119889119910)

1119902

fl 119868 + 119868119868

(68)


119868 le119862

119908119902 (119876)120581119901

sdot sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast (int

119876

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119901119908 (119910)

119901119889119910)

1119901

= 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)(

119908119902(119876)

119908119902 (119876))

120581119901

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

(69)


Ω1205721198912(119910) contained in 119868119868 By


10038161003816100381610038161003816119879119860

Ω1205721198912(119910)10038161003816100381610038161003816

le int(119876)119888

10038161003816100381610038161003816119877119898(119860119876 119910 119911)

100381610038161003816100381610038161003816100381610038161003816119910 minus 119911

1003816100381610038161003816

119898minus1

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

le 119862int(119876)119888

sum

|120574|=119898minus1

(1

10038161003816100381610038161003816119868119911119910

10038161003816100381610038161003816

int119868119911

119910

10038161003816100381610038161003816119863120574119860119876(119905)10038161003816100381610038161003816

119897

119889119905)

1119897

sdot1003816100381610038161003816Ω (119910 minus 119911)

1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572119889119911

+ 119862int(119876)119888

sum

|120574|=119898minus1

10038161003816100381610038161003816119863120574119860 (119911) minus 119898

119876(119863120574119860)10038161003816100381610038161003816

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

sdot

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572119889119911 fl 119868119868

1+ 119868119868

2

(70)

We estimate 1198681198681and 119868119868



isin 119860(1199011199041015840 119902119904

1015840) we get

1198681198681le 119862 sum

|120574|=119898minus1

(1

10038161003816100381610038161003816119868119911119910

10038161003816100381610038161003816

int119868119911

119910

10038161003816100381610038161003816119863120574119860 (119905) minus 119898

119876(119863120574119860)10038161003816100381610038161003816

119897

119889119905)

1119897

sdot

infin

sum

119895=1

int2119895+11198762119895119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572119889119911

le 119862 sum

|120574|=119898minus1

[

[

(1

10038161003816100381610038161003816119868119911119910

10038161003816100381610038161003816

sdot int119868119911

119910

100381610038161003816100381610038161003816119863120574119860 (119905) minus 119898119868119911

119910

(119863120574119860)100381610038161003816100381610038161003816

119897

119889119905)

1119897

+100381610038161003816100381610038161003816119898119868119911

119910

(119863120574119860)

minus 1198985119899119876(119863120574119860)100381610038161003816100381610038161003816+100381610038161003816100381610038161198985119899119876(119863120574119860) minus 119898

119876(119863120574119860)10038161003816100381610038161003816]

]

sdot

infin

sum

119895=1

(int2119895+1119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

119904119889119911)

1119904

sdot (int2119895+11198762119895119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

(119899minus120572)1199041015840119889119911)

11199041015840

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast Ω119871

119904(119878119899minus1)

sdot

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus120572119899(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840


le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus120572119899(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

119901

sdot 119908 (119911)119901119889119911)

1119901

sdot (int2119895+1119876

119908 (119911)minus1199011199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

sdot

infin

sum

119895=1

1

119908119902 (2119895+1119876)1119902minus120581119901

(71)


1198681198682le 119862

infin

sum

119895=1

int2119895+11198762119895119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

sdot1003816100381610038161003816Ω (119910 minus 119911)

1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572119889119911

+ 119862

infin

sum

119895=1

int2119895+11198762119895119876

sum

|120574|=119898minus1

100381610038161003816100381610038161198982119895+1119876(119863120574119860) minus 119898

119876(119863120574119860)10038161003816100381610038161003816

sdot1003816100381610038161003816Ω (119910 minus 119911)

1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572119889119911 fl 119868119868

21+ 119868119868

22

(72)


11986811986821le 119862

infin

sum

119895=1

1

100381610038161003816100381621198951198761003816100381610038161003816

1minus

120572

119899

int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863

120574119860)10038161003816100381610038161003816100381610038161003816Ω (119910 minus 119911)

10038161003816100381610038161003816100381610038161003816119891 (119911)

1003816100381610038161003816 119889119911

le 119862

infin

sum

119895=1

1

100381610038161003816100381621198951198761003816100381610038161003816

1minus

120572

119899

(int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

1199041015840

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840

(int2119895+1119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

119904119889119911)

1119904

le 119862 Ω119871119904(119878119899minus1)

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus

120572

119899

(int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

1199041015840

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840

le 119862 Ω119871119904(119878119899minus1)

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus

120572

119899

(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

119901119908 (119911)

119901119889119911)

1119901

sdot (int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

1199011199041015840

(119901minus1199041015840

)119908 (119911)

minus1199011199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

(73)


(int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

1199011199041015840

(119901minus1199041015840

)119908 (119911)

minus1199011199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

le 119862(int2119895+1119876

sum

|120574|=119898minus1

10038161003816100381610038161003816119863120574119860 (119911) minus 119898

2119895+1119876119908minus1199011199041015840(119901minus1199041015840)(119863120574119860)10038161003816100381610038161003816

1199011199041015840

(119901minus1199041015840

)

119908 (119911)minus1199011199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

+ sum

|120574|=119898minus1


2119895+1119876(119863120574119860)10038161003816100381610038161003816119908minus1199011199041015840

(119901minus1199041015840

)(2119895+1119876)(119901minus1199041015840

)1199011199041015840

fl 119868119868119868 + 119868119881

(74)


isin 119860(1199011199041015840 119902119904


119908minus1199011199041015840

(119901minus1199041015840

)isin 119860

1199051015840 sub 119860



(119901minus1199041015840


of BMO(R119899) and 1199081199041015840

isin 119860(1199011199041015840 119902119904

1015840) we have


119868119868119868 le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast 119908

minus1199011199041015840

(119901minus1199041015840

)(2119895+1119876)(119901minus1199041015840

)1199011199041015840

= 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

100381610038161003816100381610038162119895+111987610038161003816100381610038161003816

(119901minus1199041015840

)1199011199041015840

+1119902

119908119902 (2119895+1119876)1119902

(75)



119905 isin 2119895+1119876 sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816 gt 119904

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le 1198621

100381610038161003816100381610038162119895+111987610038161003816100381610038161003816119890minus1198622119904(sum|120574|=119898minus1

119863120574

119860lowast)

(76)



119908(

119905 isin 2119895+1119876 sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

gt 119904

) le 119862119908(2119895+1119876) 119890

minus1198622119904120575(sum

|120574|=119898minus1119863120574

119860lowast)

(77)


sum

|120574|=119898minus1


2119895+1119876(119863120574119860)10038161003816100381610038161003816

le1

119908minus1199011199041015840(119901minus119904

1015840) (2119895+1119876)

int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119905)

minus 1198982119895+1119876(119863120574119860)1003816100381610038161003816 119908minus1199011199041015840

(119901minus1199041015840

)(119905) 119889119905

=119862

119908minus1199011199041015840(119901minus119904

1015840) (2119895+1119876)

int

infin

0

119908minus1199011199041015840

(119901minus1199041015840

)(

119905

isin 2119895+1119876 sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

gt 119904

)119889119904 le119862

119908minus1199011199041015840(119901minus119904

1015840) (2119895+1119876)

sdot int

infin

0

119908minus1199011199041015840

(119901minus1199041015840

)(2119895+1119876) 119890

minus1198622119904120575(sum

|120574|=119898minus1119863120574

119860lowast)119889119904

= 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

(78)

As a result

119868119881 le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast 119908

minus1199011199041015840

(119901minus1199041015840

)(2119895+1119876)(119901minus1199041015840

)1199011199041015840

= 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

100381610038161003816100381610038162119895+111987610038161003816100381610038161003816

(119901minus1199041015840

)1199011199041015840

+1119902

119908119902 (2119895+1119876)1119902

(79)

Thus

11986811986821le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

sdot

infin

sum

119895=1

1

119908119902 (2119895+1119876)1119902minus120581119901

(80)


1199081199041015840

isin 119860(1199011199041015840 119902119904

1015840) we get

11986811986822le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

sdot

infin

sum

119895=1

119895 int2119895+11198762119895119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572119889119911

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast Ω119871

119904(119878119899minus1)

infin

sum

119895=1

119895

sdot

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus120572119899(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

infin

sum

119895=1

119895

sdot

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus120572119899(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

119901119908 (119911)

119901119889119911)

1119901

sdot (int2119895+1119876

119908 (119911)minus1199011199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

sdot

infin

sum

119895=1

119895

119908119902 (2119895+1119876)1119902minus120581119901

(81)

Hence

10038161003816100381610038161003816119879119860

Ω1205721198912(119910)10038161003816100381610038161003816le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

sdot

infin

sum

119895=1

119895

119908119902 (2119895+1119876)1119902minus120581119901

(82)

Therefore

119868119868 le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

sdot

infin

sum

119895=1

119895119908119902(119876)

1119902minus120581119901

119908119902 (2119895+1119876)1119902minus120581119901


le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

infin

sum

119895=1

119895

(119863119895+1

1)1119902minus120581119901

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

(83)


1

119908119902 (119876)120581119901(int119876

10038161003816100381610038161003816119879119860

Ω120572119891 (119910)

10038161003816100381610038161003816

119902

119908 (119910)119902119889119910)

1119902

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

(84)




119876+ 119891120594

(119876)119888 fl 119891

1+ 119891

2 Then we have

1

119908 (119876)120581119901(int119876

1003816100381610038161003816[119860 119879Ω] 119891 (119910)1003816100381610038161003816

119901119908 (119910) 119889119910)

1119901

le1

119908 (119876)120581119901(int119876

1003816100381610038161003816[119860 119879Ω] 1198911 (119910)1003816100381610038161003816

119901119908 (119910) 119889119910)

1119901

+1

119908 (119876)120581119901(int119876

1003816100381610038161003816[119860 119879Ω] 1198912 (119910)1003816100381610038161003816

119901119908 (119910) 119889119910)

1119901

fl 119868 + 119868119868

(85)


119868 le1

119908 (119876)120581119901

1003817100381710038171003817[119860 119879Ω] 11989111003817100381710038171003817119871119901(119908)

le119862

119908 (119876)120581119901119860lowast

100381710038171003817100381711989111003817100381710038171003817119871119901(119908)

= 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

119908(119876)120581119901

119908 (119876)120581119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(86)


1003816100381610038161003816[119860 119879Ω] 1198912 (119910)1003816100381610038161003816 le

infin

sum

119895=1

int2119895+11198762119895119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899

1003816100381610038161003816119860 (119910) minus 119860 (119911)1003816100381610038161003816

sdot1003816100381610038161003816119891 (119911)

1003816100381610038161003816 119889119911

le 119862

infin

sum

119895=1

1

100381610038161003816100381621198951198761003816100381610038161003816

(int2119895+1119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

119904119889119911)

1119904

sdot (int2119895+1119876

1003816100381610038161003816119860 (119910) minus 119860 (119911)1003816100381610038161003816

1199041015840

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840

le 119862 Ω119871119904

sdot

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

sdot (int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840

+ 119862 Ω119871119904

sdot

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

(int2119895+1119876


10038161003816100381610038161003816

1199041015840

sdot1003816100381610038161003816119891 (119911)

1003816100381610038161003816

1199041015840

119889119911)

11199041015840

fl 1198681198681(119910) + 119868119868

2

(87)




1

119908 (119876)120581119901(int119876

1198681198681(119910)

119901119908 (119910) 119889119910)

1119901

= 119862Ω119871119904

119908 (119876)120581119901

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

(int119876

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

119901

(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

1199011199041015840

119908 (119910) 119889119910)

1119901

le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

119908 (119876)

120581

119901

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

119908 (2119895+1119876)120581119901

119908minus1199041015840

(119901minus1199041015840

)(2119895+1119876)(119901minus1199041015840

)1199011199041015840

sdot (int119876

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

119908 (119876)120581119901

infin

sum

119895=1

1

119908 (2119895+1119876)(1minus120581)119901

(int119876

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

(88)



(int119876

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

le (int119876

1003816100381610038161003816119860 (119910) minus 119898119876119908 (119860)1003816100381610038161003816

119901119908 (119910) 119889119910)

1119901

+10038161003816100381610038161003816119898119876119908 (119860) minus 1198982119895+1119876119908minus119904

1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816119908 (119876)

1119901

fl 119868119868119868 + 119868119881

(89)



Lemma 25 that

119868119868119868 le 119860lowast 119908 (119876)1119901 (90)


10038161003816100381610038161003816119898119876119908 (119860) minus 1198982119895+1119876119908minus119904

1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

le1003816100381610038161003816119898119876119908 (119860) minus 119898119876 (119860)

1003816100381610038161003816 +1003816100381610038161003816119898119876 (119860) minus 1198982119895+1119876 (119860)

1003816100381610038161003816

+100381610038161003816100381610038161198982119895+1119876 (119860) minus 1198982119895+1119876119908minus119904

1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816le

1

119908 (119876)

sdot int119876

1003816100381610038161003816119860 (119905) minus 119898119876 (119860)1003816100381610038161003816 119908 (119905) 119889119905 + 2

119899(119895 + 1) 119860lowast

+1

119908minus1199041015840(119901minus119904

1015840) (2119895+1119876)

sdot int2119895+1119876

1003816100381610038161003816119860 (119905) minus 1198982119895+1119876 (119860)1003816100381610038161003816 119908minus1199041015840

(119901minus1199041015840

)(119905) 119889119905

le 119862 (119895 + 1) 119860lowast

(91)

Hence

119868119881 le 119862 (119895 + 1) 119860lowast 119908 (119876)1119901 (92)

As a result

1

119908 (119876)120581119901(int119876

1198681198681(119910)

119901119908 (119910) 119889119910)

1119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

infin

sum

119895=1

(119895 + 1)119908 (119876)

(1minus120581)119901

119908 (2119895+1119876)(1minus120581)119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(93)


1199011199041015840 we get

1198681198682le 119862 Ω119871119904

sdot

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

119901119908 (119911) 119889119911)

1119901

sdot (int2119895+1119876


10038161003816100381610038161003816

1199011199041015840

(119901minus1199041015840

)

sdot 119908 (119911)minus1199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

le 119862 Ω119871119904 119860lowast

sdot10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

119908 (2119895+1119876)120581119901

sdot 119908minus1199041015840

(119901minus1199041015840

)(2119895+1119876)(119901minus1199041015840

)1199011199041015840

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

sdot

infin

sum

119895=1

1

119908 (2119895+1119876)(1minus120581)119901

(94)

Therefore

1

119908 (119876)120581119901(int119876

119868119868119901

2119908 (119910) 119889119910)

1119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

infin

sum

119895=1

119908 (119876)(1minus120581)119901

119908 (2119895+1119876)(1minus120581)119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(95)




119860119876(119910) = 119860 (119910) minus 119898

119876(nabla119860) 119910 (96)


(119876)119888 fl

1198911+ 119891

2 Then we get

1

119908 (119876)120581119901(int119876

100381610038161003816100381610038161003816119860

Ω119891 (119910)

100381610038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

le1

119908 (119876)120581119901(int119876

100381610038161003816100381610038161003816119860

Ω1198911(119910)100381610038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

+1

119908 (119876)120581119901(int119876

100381610038161003816100381610038161003816119860

Ω1198912(119910)100381610038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

fl 119868 + 119868119868

(97)



119868 le1

119908 (119876)120581119901

100381710038171003817100381710038171003817119860

Ω1198911

100381710038171003817100381710038171003817119871119901(119908)

le119862


100381710038171003817100381711989111003817100381710038171003817119871119901(119908)

le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

119908(119876)120581119901

119908 (119876)120581119901

le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(98)


119901 Ω isin 119871




Competing Interests


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











2

100381610038161003816100381610038161198871198761

minus 1198871198762

10038161003816100381610038161003816le 119862 log(

100381610038161003816100381611987621003816100381610038161003816

100381610038161003816100381611987611003816100381610038161003816

) 119887lowast (36)


1003816100381610038161003816 le 2119899(119895 + 1) 119887lowast (37)



119878119899minus1Ω(119909

1015840)119889120590(119909

1015840) = 0 If 119887 isin BMO(R119899) then [119887 119879

Ω]




isin 11986011990110158401199041015840

(c) 1 le 119901 lt infin and 1199081199041015840

isin 119860119901




Ω119891100381710038171003817100381710038171003817119871119901(119908)




2 More


119908 (120582119876) le 119862120582119899119901119908 (119876) (39)



119908 (119878)

119908 (119876)le 119862(

|119878|

|119876|)

120575

(40)



BMO (119908) = 119887 119887lowast119908

= sup119876

1

119908 (119876)int119876

1003816100381610038161003816119887 (119909) minus 1198981198761199081198871003816100381610038161003816 119908 (119909) 119889119909

119898119876119908119887 =

1

119908 (119876)int119876

119887 (119909)119908 (119909) 119889119909

(41)



Ω120572119891(119909) at first Set

119879Ω120572+120573

119891 (119909) = intR119899

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572minus120573

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

0 lt 120572 + 120573 lt 119899

(42)




10038161003816100381610038161003816119879119860

Ω120572119891 (119909)

10038161003816100381610038161003816le 119862( sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573

)119879Ω120572+120573

119891 (119909) (43)


119896= 2

119896119876 and set

119860119876119896

(119910) = 119860 (119910) minus sum

|120574|=119898minus1

1

120574119898119876119896

(119863120574119860)119910

120574 (44)

where 119898119876119896


|120574| = 119898 minus 1

119863120574119860119876119896

(119910) = 119863120574119860 (119910) minus 119898

119876119896

(119863120574119860) (45)


119877119898(119860 119909 119910) = 119877

119898(119860119876119896

119909 119910) (46)

Hence

10038161003816100381610038161003816119879119860

Ω120572119891 (119909)

10038161003816100381610038161003816

le

infin

sum

119896=minusinfin


119896119903

10038161003816100381610038161003816119877119898(119860119876119896

119909 119910)10038161003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119898minus1

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

flinfin

sum

119896=minusinfin

119879119896

(47)

By Lemma 15 we get

10038161003816100381610038161003816119877119898(119860119876119896

119909 119910)10038161003816100381610038161003816

le10038161003816100381610038161003816119877119898minus1

(119860119876119896

119909 119910)10038161003816100381610038161003816

+ 119862 sum

|120574|=119898minus1

10038161003816100381610038161003816119863120574119860119876119896

(119910)10038161003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119898minus1

le 1198621003816100381610038161003816119909 minus 119910

1003816100381610038161003816

119898minus1sum

|120574|=119898minus1

(1

1003816100381610038161003816119868119910

119909

1003816100381610038161003816

int119868119910

119909

10038161003816100381610038161003816119863120574119860119876119896

(119911)10038161003816100381610038161003816

119897

119889119911)

1119897

+ 1198621003816100381610038161003816119909 minus 119910

1003816100381610038161003816

119898minus1sum

|120574|=119898minus1

10038161003816100381610038161003816119863120574119860119876119896

(119910)10038161003816100381610038161003816

(48)



119896 By Lemmas 16


(1

1003816100381610038161003816119868119910

119909

1003816100381610038161003816

int119868119910

119909

10038161003816100381610038161003816119863120574119860119876119896

(119911)10038161003816100381610038161003816

119897

119889119911)

1119897

= (1

1003816100381610038161003816119868119910

119909

1003816100381610038161003816

int119868119910

119909

10038161003816100381610038161003816119863120574119860 (119911) minus 119898119876

119896

(119863120574119860)10038161003816100381610038161003816

119897

119889119911)

1119897

le (1

1003816100381610038161003816119868119910

119909

1003816100381610038161003816

int119868119910

119909

10038161003816100381610038161003816119863120574119860 (119911) minus 119898

119868119910

119909

(119863120574119860)10038161003816100381610038161003816

119897

119889119911)

1119897

+10038161003816100381610038161003816119898119868119910

119909

(119863120574119860) minus 119898

5119899119876119896

(119863120574119860)10038161003816100381610038161003816

+100381610038161003816100381610038161198985119899119876119896

(119863120574119860) minus 119898

119876119896

(119863120574119860)10038161003816100381610038161003816

le 11986210038161003816100381610038161198761198961003816100381610038161003816

120573119899 10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573

le 119862 (2119896119903)120573 1003817100381710038171003817119863

1205741198601003817100381710038171003817Λ120573

(49)


10038161003816100381610038161003816119863120574119860119876119896

(119910)10038161003816100381610038161003816=10038161003816100381610038161003816119863120574119860 (119910) minus 119898

119876119896

(119863120574119860)10038161003816100381610038161003816

le 11986210038161003816100381610038161198761198961003816100381610038161003816

120573119899 10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573

le 119862 (2119896119903)120573 1003817100381710038171003817119863

1205741198601003817100381710038171003817Λ120573

(50)

Thus

10038161003816100381610038161003816119877119898(119860119876119896

119909 119910)10038161003816100381610038161003816

le 1198621003816100381610038161003816119909 minus 119910

1003816100381610038161003816

119898minus1(2119896119903)120573

sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573

(51)

Therefore

119879119896le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573


119896119903

(2119896119903)120573

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572

1003816100381610038161003816Ω (119909 minus 119910)10038161003816100381610038161003816100381610038161003816119891 (119910)

1003816100381610038161003816 119889119910

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573


119896119903

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572minus120573

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

(52)

It follows that

10038161003816100381610038161003816119879119860

Ω120572119891 (119909)

10038161003816100381610038161003816le

infin

sum

119896=minusinfin

(119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573


119896119903

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572minus120573

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910)

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573

sdot

infin

sum

119896=minusinfin


119896119903

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572minus120573

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

= 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573

intR119899

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572minus120573

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

= 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573

119879Ω120572+120573

119891 (119909)

(53)






1+ 119891

2with

1198911= 119891120594


(1

119908119902 (119861)120581119902119901

int119861

10038161003816100381610038161003816119879Ω120572+120573

119891 (119909)10038161003816100381610038161003816

119902

119908119902(119909) 119889119909)

1119902

le1

119908119902 (119861)120581119901(int119861

10038161003816100381610038161003816119879Ω120572+120573

1198911 (119909)

10038161003816100381610038161003816

119902

119908119902(119909) 119889119909)

1119902

+1

119908119902 (119861)120581119901(int119861

10038161003816100381610038161003816119879Ω120572+120573

1198912 (119909)

10038161003816100381610038161003816

119902

119908119902(119909) 119889119909)

1119902

fl 1198691+ 1198692

(54)


2


1198691le

1

119908119902 (119861)120581119901

10038171003817100381710038171003817119879Ω120572+120573

1198911

10038171003817100381710038171003817119871119902(119908119902)

le119862

119908119902 (119861)120581119901

100381710038171003817100381711989111003817100381710038171003817119871119901(119908119901)

=119862

119908119902 (119861)120581119901(int2119861

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119901119908 (119909)

119901119889119909)

1119901

le 11986210038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

119908119902(2119861)

120581119901

119908119902 (119861)120581119901

le 11986210038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

(55)



10038161003816100381610038161003816119879Ω120572+120573

1198912 (119909)

10038161003816100381610038161003816=

infin

sum

119895=1

int2119895+11198612119895119861

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572minus120573

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

le 119862

infin

sum

119895=1

(int2119895+1119861

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

119904119889119910)

1119904

sdot (int2119895+11198612119895119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816

1199041015840

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

(119899minus120572minus120573)1199041015840119889119910)

11199041015840

fl 119862infin

sum

119895=1

(11986811198951198682119895)

(56)



1198681119895= (int

2119895+2119861

|Ω (119911)|119904119889119911)

1119904

= (int

2119895+2

119903119861

0

int119878119899minus1

10038161003816100381610038161003816Ω (119911

1015840)10038161003816100381610038161003816

119904

1198891199111015840119903119899minus1119889119903)

1119904

= 119862 Ω119871119904(119878119899minus1)

100381610038161003816100381610038162119895+211986110038161003816100381610038161003816

1119904

(57)


1198682119895le

119862

10038161003816100381610038162119895+1119861

1003816100381610038161003816

1minus(120572+120573)119899(int2119895+1119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816

1199041015840

119889119910)

11199041015840

(58)


isin 119860(1199011199041015840 119902119904

1015840) we get

(int2119895+1119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816

1199041015840

119889119910)

11199041015840

le 119862(int2119895+1119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119901119908 (119910)

119901119889119910)

1119901

sdot (int2119895+1119861

119908 (119910)minus1199011199041015840

(119901minus1199041015840

)119889119910)

(119901minus1199041015840

)1199011199041015840

le 11986210038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) 119908

119902(2119895+1119861)120581119901

sdot (int2119895+1119861

119908 (119910)minus1199011199041015840

(119901minus1199041015840

)119889119910)

(119901minus1199041015840

)1199011199041015840

le 11986210038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) 119908

119902(2119895+1119861)120581119901

sdot

100381610038161003816100381610038162119895+111986110038161003816100381610038161003816

(119901119902minus1199041015840

119902+1199041015840

119901)1199011199021199041015840

119908119902 (2119895+1119861)1119902

(59)

Thus

10038161003816100381610038161003816119879Ω120572+120573

1198912 (119909)

10038161003816100381610038161003816le 119862

infin

sum

119895=1

(11986811198951198682119895)

le 119862

infin

sum

119895=1

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

1

119908119902 (2119895+1119861)1119902minus120581119901

(60)

So we get

1198692le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

infin

sum

119895=1

119908119902(119861)

1119902minus120581119901

119908119902 (2119895+1119861)1119902minus120581119901

(61)


1198692le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) (62)



Ω120572is bounded




119879119860

Ω120572119891 (119909)

= intR119899

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572+119898minus1

1003816100381610038161003816119877119898 (119860 119909 119910)10038161003816100381610038161003816100381610038161003816119891 (119910)

1003816100381610038161003816 119889119910

0 le 120572 lt 119899

(63)


119860

Ω120572 On the


119879119860

Ω120572119891 (119909)

ge int|119909minus119910|lt119903

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572+119898minus1

1003816100381610038161003816119877119898 (119860 119909 119910)10038161003816100381610038161003816100381610038161003816119891 (119910)

1003816100381610038161003816 119889119910

ge1

119903119899minus120572+119898minus1


1003816100381610038161003816Ω (119909 minus 119910)10038161003816100381610038161003816100381610038161003816119877119898 (119860 119909 119910)

10038161003816100381610038161003816100381610038161003816119891 (119910)

1003816100381610038161003816 119889119910

(64)


119879119860

Ω120572119891 (119909) ge 119872

119860

Ω120572119891 (119909) (65)




119879Ω120573119891 (119909) = int

R119899

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120573

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910 (66)







119860119876(119910) = 119860 (119910) minus sum

|120574|=119898minus1

1

120574119898119876(119863120574119860)119910

120574 (67)


119876+ 119891120594

(119876)119888 = 119891

1+ 119891

2

Then we have

1

119908119902 (119876)120581119901(int119876

10038161003816100381610038161003816119879119860

Ω120572119891 (119910)

10038161003816100381610038161003816

119902

119908 (119910)119902119889119910)

1119902

le1

119908119902 (119876)120581119901(int119876

10038161003816100381610038161003816119879119860

Ω1205721198911(119910)10038161003816100381610038161003816

119902

119908 (119910)119902119889119910)

1119902

+1

119908119902 (119876)120581119901(int119876

10038161003816100381610038161003816119879119860

Ω1205721198912(119910)10038161003816100381610038161003816

119902

119908 (119910)119902119889119910)

1119902

fl 119868 + 119868119868

(68)


119868 le119862

119908119902 (119876)120581119901

sdot sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast (int

119876

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119901119908 (119910)

119901119889119910)

1119901

= 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)(

119908119902(119876)

119908119902 (119876))

120581119901

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

(69)


Ω1205721198912(119910) contained in 119868119868 By


10038161003816100381610038161003816119879119860

Ω1205721198912(119910)10038161003816100381610038161003816

le int(119876)119888

10038161003816100381610038161003816119877119898(119860119876 119910 119911)

100381610038161003816100381610038161003816100381610038161003816119910 minus 119911

1003816100381610038161003816

119898minus1

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

le 119862int(119876)119888

sum

|120574|=119898minus1

(1

10038161003816100381610038161003816119868119911119910

10038161003816100381610038161003816

int119868119911

119910

10038161003816100381610038161003816119863120574119860119876(119905)10038161003816100381610038161003816

119897

119889119905)

1119897

sdot1003816100381610038161003816Ω (119910 minus 119911)

1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572119889119911

+ 119862int(119876)119888

sum

|120574|=119898minus1

10038161003816100381610038161003816119863120574119860 (119911) minus 119898

119876(119863120574119860)10038161003816100381610038161003816

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

sdot

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572119889119911 fl 119868119868

1+ 119868119868

2

(70)

We estimate 1198681198681and 119868119868



isin 119860(1199011199041015840 119902119904

1015840) we get

1198681198681le 119862 sum

|120574|=119898minus1

(1

10038161003816100381610038161003816119868119911119910

10038161003816100381610038161003816

int119868119911

119910

10038161003816100381610038161003816119863120574119860 (119905) minus 119898

119876(119863120574119860)10038161003816100381610038161003816

119897

119889119905)

1119897

sdot

infin

sum

119895=1

int2119895+11198762119895119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572119889119911

le 119862 sum

|120574|=119898minus1

[

[

(1

10038161003816100381610038161003816119868119911119910

10038161003816100381610038161003816

sdot int119868119911

119910

100381610038161003816100381610038161003816119863120574119860 (119905) minus 119898119868119911

119910

(119863120574119860)100381610038161003816100381610038161003816

119897

119889119905)

1119897

+100381610038161003816100381610038161003816119898119868119911

119910

(119863120574119860)

minus 1198985119899119876(119863120574119860)100381610038161003816100381610038161003816+100381610038161003816100381610038161198985119899119876(119863120574119860) minus 119898

119876(119863120574119860)10038161003816100381610038161003816]

]

sdot

infin

sum

119895=1

(int2119895+1119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

119904119889119911)

1119904

sdot (int2119895+11198762119895119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

(119899minus120572)1199041015840119889119911)

11199041015840

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast Ω119871

119904(119878119899minus1)

sdot

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus120572119899(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840


le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus120572119899(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

119901

sdot 119908 (119911)119901119889119911)

1119901

sdot (int2119895+1119876

119908 (119911)minus1199011199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

sdot

infin

sum

119895=1

1

119908119902 (2119895+1119876)1119902minus120581119901

(71)


1198681198682le 119862

infin

sum

119895=1

int2119895+11198762119895119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

sdot1003816100381610038161003816Ω (119910 minus 119911)

1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572119889119911

+ 119862

infin

sum

119895=1

int2119895+11198762119895119876

sum

|120574|=119898minus1

100381610038161003816100381610038161198982119895+1119876(119863120574119860) minus 119898

119876(119863120574119860)10038161003816100381610038161003816

sdot1003816100381610038161003816Ω (119910 minus 119911)

1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572119889119911 fl 119868119868

21+ 119868119868

22

(72)


11986811986821le 119862

infin

sum

119895=1

1

100381610038161003816100381621198951198761003816100381610038161003816

1minus

120572

119899

int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863

120574119860)10038161003816100381610038161003816100381610038161003816Ω (119910 minus 119911)

10038161003816100381610038161003816100381610038161003816119891 (119911)

1003816100381610038161003816 119889119911

le 119862

infin

sum

119895=1

1

100381610038161003816100381621198951198761003816100381610038161003816

1minus

120572

119899

(int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

1199041015840

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840

(int2119895+1119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

119904119889119911)

1119904

le 119862 Ω119871119904(119878119899minus1)

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus

120572

119899

(int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

1199041015840

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840

le 119862 Ω119871119904(119878119899minus1)

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus

120572

119899

(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

119901119908 (119911)

119901119889119911)

1119901

sdot (int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

1199011199041015840

(119901minus1199041015840

)119908 (119911)

minus1199011199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

(73)


(int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

1199011199041015840

(119901minus1199041015840

)119908 (119911)

minus1199011199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

le 119862(int2119895+1119876

sum

|120574|=119898minus1

10038161003816100381610038161003816119863120574119860 (119911) minus 119898

2119895+1119876119908minus1199011199041015840(119901minus1199041015840)(119863120574119860)10038161003816100381610038161003816

1199011199041015840

(119901minus1199041015840

)

119908 (119911)minus1199011199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

+ sum

|120574|=119898minus1


2119895+1119876(119863120574119860)10038161003816100381610038161003816119908minus1199011199041015840

(119901minus1199041015840

)(2119895+1119876)(119901minus1199041015840

)1199011199041015840

fl 119868119868119868 + 119868119881

(74)


isin 119860(1199011199041015840 119902119904


119908minus1199011199041015840

(119901minus1199041015840

)isin 119860

1199051015840 sub 119860



(119901minus1199041015840


of BMO(R119899) and 1199081199041015840

isin 119860(1199011199041015840 119902119904

1015840) we have


119868119868119868 le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast 119908

minus1199011199041015840

(119901minus1199041015840

)(2119895+1119876)(119901minus1199041015840

)1199011199041015840

= 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

100381610038161003816100381610038162119895+111987610038161003816100381610038161003816

(119901minus1199041015840

)1199011199041015840

+1119902

119908119902 (2119895+1119876)1119902

(75)



119905 isin 2119895+1119876 sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816 gt 119904

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le 1198621

100381610038161003816100381610038162119895+111987610038161003816100381610038161003816119890minus1198622119904(sum|120574|=119898minus1

119863120574

119860lowast)

(76)



119908(

119905 isin 2119895+1119876 sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

gt 119904

) le 119862119908(2119895+1119876) 119890

minus1198622119904120575(sum

|120574|=119898minus1119863120574

119860lowast)

(77)


sum

|120574|=119898minus1


2119895+1119876(119863120574119860)10038161003816100381610038161003816

le1

119908minus1199011199041015840(119901minus119904

1015840) (2119895+1119876)

int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119905)

minus 1198982119895+1119876(119863120574119860)1003816100381610038161003816 119908minus1199011199041015840

(119901minus1199041015840

)(119905) 119889119905

=119862

119908minus1199011199041015840(119901minus119904

1015840) (2119895+1119876)

int

infin

0

119908minus1199011199041015840

(119901minus1199041015840

)(

119905

isin 2119895+1119876 sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

gt 119904

)119889119904 le119862

119908minus1199011199041015840(119901minus119904

1015840) (2119895+1119876)

sdot int

infin

0

119908minus1199011199041015840

(119901minus1199041015840

)(2119895+1119876) 119890

minus1198622119904120575(sum

|120574|=119898minus1119863120574

119860lowast)119889119904

= 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

(78)

As a result

119868119881 le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast 119908

minus1199011199041015840

(119901minus1199041015840

)(2119895+1119876)(119901minus1199041015840

)1199011199041015840

= 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

100381610038161003816100381610038162119895+111987610038161003816100381610038161003816

(119901minus1199041015840

)1199011199041015840

+1119902

119908119902 (2119895+1119876)1119902

(79)

Thus

11986811986821le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

sdot

infin

sum

119895=1

1

119908119902 (2119895+1119876)1119902minus120581119901

(80)


1199081199041015840

isin 119860(1199011199041015840 119902119904

1015840) we get

11986811986822le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

sdot

infin

sum

119895=1

119895 int2119895+11198762119895119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572119889119911

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast Ω119871

119904(119878119899minus1)

infin

sum

119895=1

119895

sdot

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus120572119899(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

infin

sum

119895=1

119895

sdot

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus120572119899(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

119901119908 (119911)

119901119889119911)

1119901

sdot (int2119895+1119876

119908 (119911)minus1199011199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

sdot

infin

sum

119895=1

119895

119908119902 (2119895+1119876)1119902minus120581119901

(81)

Hence

10038161003816100381610038161003816119879119860

Ω1205721198912(119910)10038161003816100381610038161003816le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

sdot

infin

sum

119895=1

119895

119908119902 (2119895+1119876)1119902minus120581119901

(82)

Therefore

119868119868 le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

sdot

infin

sum

119895=1

119895119908119902(119876)

1119902minus120581119901

119908119902 (2119895+1119876)1119902minus120581119901


le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

infin

sum

119895=1

119895

(119863119895+1

1)1119902minus120581119901

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

(83)


1

119908119902 (119876)120581119901(int119876

10038161003816100381610038161003816119879119860

Ω120572119891 (119910)

10038161003816100381610038161003816

119902

119908 (119910)119902119889119910)

1119902

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

(84)




119876+ 119891120594

(119876)119888 fl 119891

1+ 119891

2 Then we have

1

119908 (119876)120581119901(int119876

1003816100381610038161003816[119860 119879Ω] 119891 (119910)1003816100381610038161003816

119901119908 (119910) 119889119910)

1119901

le1

119908 (119876)120581119901(int119876

1003816100381610038161003816[119860 119879Ω] 1198911 (119910)1003816100381610038161003816

119901119908 (119910) 119889119910)

1119901

+1

119908 (119876)120581119901(int119876

1003816100381610038161003816[119860 119879Ω] 1198912 (119910)1003816100381610038161003816

119901119908 (119910) 119889119910)

1119901

fl 119868 + 119868119868

(85)


119868 le1

119908 (119876)120581119901

1003817100381710038171003817[119860 119879Ω] 11989111003817100381710038171003817119871119901(119908)

le119862

119908 (119876)120581119901119860lowast

100381710038171003817100381711989111003817100381710038171003817119871119901(119908)

= 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

119908(119876)120581119901

119908 (119876)120581119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(86)


1003816100381610038161003816[119860 119879Ω] 1198912 (119910)1003816100381610038161003816 le

infin

sum

119895=1

int2119895+11198762119895119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899

1003816100381610038161003816119860 (119910) minus 119860 (119911)1003816100381610038161003816

sdot1003816100381610038161003816119891 (119911)

1003816100381610038161003816 119889119911

le 119862

infin

sum

119895=1

1

100381610038161003816100381621198951198761003816100381610038161003816

(int2119895+1119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

119904119889119911)

1119904

sdot (int2119895+1119876

1003816100381610038161003816119860 (119910) minus 119860 (119911)1003816100381610038161003816

1199041015840

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840

le 119862 Ω119871119904

sdot

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

sdot (int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840

+ 119862 Ω119871119904

sdot

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

(int2119895+1119876


10038161003816100381610038161003816

1199041015840

sdot1003816100381610038161003816119891 (119911)

1003816100381610038161003816

1199041015840

119889119911)

11199041015840

fl 1198681198681(119910) + 119868119868

2

(87)




1

119908 (119876)120581119901(int119876

1198681198681(119910)

119901119908 (119910) 119889119910)

1119901

= 119862Ω119871119904

119908 (119876)120581119901

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

(int119876

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

119901

(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

1199011199041015840

119908 (119910) 119889119910)

1119901

le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

119908 (119876)

120581

119901

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

119908 (2119895+1119876)120581119901

119908minus1199041015840

(119901minus1199041015840

)(2119895+1119876)(119901minus1199041015840

)1199011199041015840

sdot (int119876

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

119908 (119876)120581119901

infin

sum

119895=1

1

119908 (2119895+1119876)(1minus120581)119901

(int119876

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

(88)



(int119876

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

le (int119876

1003816100381610038161003816119860 (119910) minus 119898119876119908 (119860)1003816100381610038161003816

119901119908 (119910) 119889119910)

1119901

+10038161003816100381610038161003816119898119876119908 (119860) minus 1198982119895+1119876119908minus119904

1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816119908 (119876)

1119901

fl 119868119868119868 + 119868119881

(89)



Lemma 25 that

119868119868119868 le 119860lowast 119908 (119876)1119901 (90)


10038161003816100381610038161003816119898119876119908 (119860) minus 1198982119895+1119876119908minus119904

1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

le1003816100381610038161003816119898119876119908 (119860) minus 119898119876 (119860)

1003816100381610038161003816 +1003816100381610038161003816119898119876 (119860) minus 1198982119895+1119876 (119860)

1003816100381610038161003816

+100381610038161003816100381610038161198982119895+1119876 (119860) minus 1198982119895+1119876119908minus119904

1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816le

1

119908 (119876)

sdot int119876

1003816100381610038161003816119860 (119905) minus 119898119876 (119860)1003816100381610038161003816 119908 (119905) 119889119905 + 2

119899(119895 + 1) 119860lowast

+1

119908minus1199041015840(119901minus119904

1015840) (2119895+1119876)

sdot int2119895+1119876

1003816100381610038161003816119860 (119905) minus 1198982119895+1119876 (119860)1003816100381610038161003816 119908minus1199041015840

(119901minus1199041015840

)(119905) 119889119905

le 119862 (119895 + 1) 119860lowast

(91)

Hence

119868119881 le 119862 (119895 + 1) 119860lowast 119908 (119876)1119901 (92)

As a result

1

119908 (119876)120581119901(int119876

1198681198681(119910)

119901119908 (119910) 119889119910)

1119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

infin

sum

119895=1

(119895 + 1)119908 (119876)

(1minus120581)119901

119908 (2119895+1119876)(1minus120581)119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(93)


1199011199041015840 we get

1198681198682le 119862 Ω119871119904

sdot

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

119901119908 (119911) 119889119911)

1119901

sdot (int2119895+1119876


10038161003816100381610038161003816

1199011199041015840

(119901minus1199041015840

)

sdot 119908 (119911)minus1199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

le 119862 Ω119871119904 119860lowast

sdot10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

119908 (2119895+1119876)120581119901

sdot 119908minus1199041015840

(119901minus1199041015840

)(2119895+1119876)(119901minus1199041015840

)1199011199041015840

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

sdot

infin

sum

119895=1

1

119908 (2119895+1119876)(1minus120581)119901

(94)

Therefore

1

119908 (119876)120581119901(int119876

119868119868119901

2119908 (119910) 119889119910)

1119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

infin

sum

119895=1

119908 (119876)(1minus120581)119901

119908 (2119895+1119876)(1minus120581)119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(95)




119860119876(119910) = 119860 (119910) minus 119898

119876(nabla119860) 119910 (96)


(119876)119888 fl

1198911+ 119891

2 Then we get

1

119908 (119876)120581119901(int119876

100381610038161003816100381610038161003816119860

Ω119891 (119910)

100381610038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

le1

119908 (119876)120581119901(int119876

100381610038161003816100381610038161003816119860

Ω1198911(119910)100381610038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

+1

119908 (119876)120581119901(int119876

100381610038161003816100381610038161003816119860

Ω1198912(119910)100381610038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

fl 119868 + 119868119868

(97)



119868 le1

119908 (119876)120581119901

100381710038171003817100381710038171003817119860

Ω1198911

100381710038171003817100381710038171003817119871119901(119908)

le119862


100381710038171003817100381711989111003817100381710038171003817119871119901(119908)

le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

119908(119876)120581119901

119908 (119876)120581119901

le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(98)


119901 Ω isin 119871




Competing Interests


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119896 By Lemmas 16


(1

1003816100381610038161003816119868119910

119909

1003816100381610038161003816

int119868119910

119909

10038161003816100381610038161003816119863120574119860119876119896

(119911)10038161003816100381610038161003816

119897

119889119911)

1119897

= (1

1003816100381610038161003816119868119910

119909

1003816100381610038161003816

int119868119910

119909

10038161003816100381610038161003816119863120574119860 (119911) minus 119898119876

119896

(119863120574119860)10038161003816100381610038161003816

119897

119889119911)

1119897

le (1

1003816100381610038161003816119868119910

119909

1003816100381610038161003816

int119868119910

119909

10038161003816100381610038161003816119863120574119860 (119911) minus 119898

119868119910

119909

(119863120574119860)10038161003816100381610038161003816

119897

119889119911)

1119897

+10038161003816100381610038161003816119898119868119910

119909

(119863120574119860) minus 119898

5119899119876119896

(119863120574119860)10038161003816100381610038161003816

+100381610038161003816100381610038161198985119899119876119896

(119863120574119860) minus 119898

119876119896

(119863120574119860)10038161003816100381610038161003816

le 11986210038161003816100381610038161198761198961003816100381610038161003816

120573119899 10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573

le 119862 (2119896119903)120573 1003817100381710038171003817119863

1205741198601003817100381710038171003817Λ120573

(49)


10038161003816100381610038161003816119863120574119860119876119896

(119910)10038161003816100381610038161003816=10038161003816100381610038161003816119863120574119860 (119910) minus 119898

119876119896

(119863120574119860)10038161003816100381610038161003816

le 11986210038161003816100381610038161198761198961003816100381610038161003816

120573119899 10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573

le 119862 (2119896119903)120573 1003817100381710038171003817119863

1205741198601003817100381710038171003817Λ120573

(50)

Thus

10038161003816100381610038161003816119877119898(119860119876119896

119909 119910)10038161003816100381610038161003816

le 1198621003816100381610038161003816119909 minus 119910

1003816100381610038161003816

119898minus1(2119896119903)120573

sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573

(51)

Therefore

119879119896le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573


119896119903

(2119896119903)120573

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572

1003816100381610038161003816Ω (119909 minus 119910)10038161003816100381610038161003816100381610038161003816119891 (119910)

1003816100381610038161003816 119889119910

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573


119896119903

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572minus120573

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

(52)

It follows that

10038161003816100381610038161003816119879119860

Ω120572119891 (119909)

10038161003816100381610038161003816le

infin

sum

119896=minusinfin

(119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573


119896119903

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572minus120573

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910)

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573

sdot

infin

sum

119896=minusinfin


119896119903

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572minus120573

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

= 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573

intR119899

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572minus120573

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

= 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573

119879Ω120572+120573

119891 (119909)

(53)






1+ 119891

2with

1198911= 119891120594


(1

119908119902 (119861)120581119902119901

int119861

10038161003816100381610038161003816119879Ω120572+120573

119891 (119909)10038161003816100381610038161003816

119902

119908119902(119909) 119889119909)

1119902

le1

119908119902 (119861)120581119901(int119861

10038161003816100381610038161003816119879Ω120572+120573

1198911 (119909)

10038161003816100381610038161003816

119902

119908119902(119909) 119889119909)

1119902

+1

119908119902 (119861)120581119901(int119861

10038161003816100381610038161003816119879Ω120572+120573

1198912 (119909)

10038161003816100381610038161003816

119902

119908119902(119909) 119889119909)

1119902

fl 1198691+ 1198692

(54)


2


1198691le

1

119908119902 (119861)120581119901

10038171003817100381710038171003817119879Ω120572+120573

1198911

10038171003817100381710038171003817119871119902(119908119902)

le119862

119908119902 (119861)120581119901

100381710038171003817100381711989111003817100381710038171003817119871119901(119908119901)

=119862

119908119902 (119861)120581119901(int2119861

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119901119908 (119909)

119901119889119909)

1119901

le 11986210038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

119908119902(2119861)

120581119901

119908119902 (119861)120581119901

le 11986210038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

(55)



10038161003816100381610038161003816119879Ω120572+120573

1198912 (119909)

10038161003816100381610038161003816=

infin

sum

119895=1

int2119895+11198612119895119861

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572minus120573

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

le 119862

infin

sum

119895=1

(int2119895+1119861

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

119904119889119910)

1119904

sdot (int2119895+11198612119895119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816

1199041015840

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

(119899minus120572minus120573)1199041015840119889119910)

11199041015840

fl 119862infin

sum

119895=1

(11986811198951198682119895)

(56)



1198681119895= (int

2119895+2119861

|Ω (119911)|119904119889119911)

1119904

= (int

2119895+2

119903119861

0

int119878119899minus1

10038161003816100381610038161003816Ω (119911

1015840)10038161003816100381610038161003816

119904

1198891199111015840119903119899minus1119889119903)

1119904

= 119862 Ω119871119904(119878119899minus1)

100381610038161003816100381610038162119895+211986110038161003816100381610038161003816

1119904

(57)


1198682119895le

119862

10038161003816100381610038162119895+1119861

1003816100381610038161003816

1minus(120572+120573)119899(int2119895+1119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816

1199041015840

119889119910)

11199041015840

(58)


isin 119860(1199011199041015840 119902119904

1015840) we get

(int2119895+1119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816

1199041015840

119889119910)

11199041015840

le 119862(int2119895+1119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119901119908 (119910)

119901119889119910)

1119901

sdot (int2119895+1119861

119908 (119910)minus1199011199041015840

(119901minus1199041015840

)119889119910)

(119901minus1199041015840

)1199011199041015840

le 11986210038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) 119908

119902(2119895+1119861)120581119901

sdot (int2119895+1119861

119908 (119910)minus1199011199041015840

(119901minus1199041015840

)119889119910)

(119901minus1199041015840

)1199011199041015840

le 11986210038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) 119908

119902(2119895+1119861)120581119901

sdot

100381610038161003816100381610038162119895+111986110038161003816100381610038161003816

(119901119902minus1199041015840

119902+1199041015840

119901)1199011199021199041015840

119908119902 (2119895+1119861)1119902

(59)

Thus

10038161003816100381610038161003816119879Ω120572+120573

1198912 (119909)

10038161003816100381610038161003816le 119862

infin

sum

119895=1

(11986811198951198682119895)

le 119862

infin

sum

119895=1

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

1

119908119902 (2119895+1119861)1119902minus120581119901

(60)

So we get

1198692le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

infin

sum

119895=1

119908119902(119861)

1119902minus120581119901

119908119902 (2119895+1119861)1119902minus120581119901

(61)


1198692le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) (62)



Ω120572is bounded




119879119860

Ω120572119891 (119909)

= intR119899

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572+119898minus1

1003816100381610038161003816119877119898 (119860 119909 119910)10038161003816100381610038161003816100381610038161003816119891 (119910)

1003816100381610038161003816 119889119910

0 le 120572 lt 119899

(63)


119860

Ω120572 On the


119879119860

Ω120572119891 (119909)

ge int|119909minus119910|lt119903

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572+119898minus1

1003816100381610038161003816119877119898 (119860 119909 119910)10038161003816100381610038161003816100381610038161003816119891 (119910)

1003816100381610038161003816 119889119910

ge1

119903119899minus120572+119898minus1


1003816100381610038161003816Ω (119909 minus 119910)10038161003816100381610038161003816100381610038161003816119877119898 (119860 119909 119910)

10038161003816100381610038161003816100381610038161003816119891 (119910)

1003816100381610038161003816 119889119910

(64)


119879119860

Ω120572119891 (119909) ge 119872

119860

Ω120572119891 (119909) (65)




119879Ω120573119891 (119909) = int

R119899

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120573

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910 (66)







119860119876(119910) = 119860 (119910) minus sum

|120574|=119898minus1

1

120574119898119876(119863120574119860)119910

120574 (67)


119876+ 119891120594

(119876)119888 = 119891

1+ 119891

2

Then we have

1

119908119902 (119876)120581119901(int119876

10038161003816100381610038161003816119879119860

Ω120572119891 (119910)

10038161003816100381610038161003816

119902

119908 (119910)119902119889119910)

1119902

le1

119908119902 (119876)120581119901(int119876

10038161003816100381610038161003816119879119860

Ω1205721198911(119910)10038161003816100381610038161003816

119902

119908 (119910)119902119889119910)

1119902

+1

119908119902 (119876)120581119901(int119876

10038161003816100381610038161003816119879119860

Ω1205721198912(119910)10038161003816100381610038161003816

119902

119908 (119910)119902119889119910)

1119902

fl 119868 + 119868119868

(68)


119868 le119862

119908119902 (119876)120581119901

sdot sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast (int

119876

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119901119908 (119910)

119901119889119910)

1119901

= 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)(

119908119902(119876)

119908119902 (119876))

120581119901

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

(69)


Ω1205721198912(119910) contained in 119868119868 By


10038161003816100381610038161003816119879119860

Ω1205721198912(119910)10038161003816100381610038161003816

le int(119876)119888

10038161003816100381610038161003816119877119898(119860119876 119910 119911)

100381610038161003816100381610038161003816100381610038161003816119910 minus 119911

1003816100381610038161003816

119898minus1

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

le 119862int(119876)119888

sum

|120574|=119898minus1

(1

10038161003816100381610038161003816119868119911119910

10038161003816100381610038161003816

int119868119911

119910

10038161003816100381610038161003816119863120574119860119876(119905)10038161003816100381610038161003816

119897

119889119905)

1119897

sdot1003816100381610038161003816Ω (119910 minus 119911)

1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572119889119911

+ 119862int(119876)119888

sum

|120574|=119898minus1

10038161003816100381610038161003816119863120574119860 (119911) minus 119898

119876(119863120574119860)10038161003816100381610038161003816

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

sdot

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572119889119911 fl 119868119868

1+ 119868119868

2

(70)

We estimate 1198681198681and 119868119868



isin 119860(1199011199041015840 119902119904

1015840) we get

1198681198681le 119862 sum

|120574|=119898minus1

(1

10038161003816100381610038161003816119868119911119910

10038161003816100381610038161003816

int119868119911

119910

10038161003816100381610038161003816119863120574119860 (119905) minus 119898

119876(119863120574119860)10038161003816100381610038161003816

119897

119889119905)

1119897

sdot

infin

sum

119895=1

int2119895+11198762119895119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572119889119911

le 119862 sum

|120574|=119898minus1

[

[

(1

10038161003816100381610038161003816119868119911119910

10038161003816100381610038161003816

sdot int119868119911

119910

100381610038161003816100381610038161003816119863120574119860 (119905) minus 119898119868119911

119910

(119863120574119860)100381610038161003816100381610038161003816

119897

119889119905)

1119897

+100381610038161003816100381610038161003816119898119868119911

119910

(119863120574119860)

minus 1198985119899119876(119863120574119860)100381610038161003816100381610038161003816+100381610038161003816100381610038161198985119899119876(119863120574119860) minus 119898

119876(119863120574119860)10038161003816100381610038161003816]

]

sdot

infin

sum

119895=1

(int2119895+1119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

119904119889119911)

1119904

sdot (int2119895+11198762119895119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

(119899minus120572)1199041015840119889119911)

11199041015840

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast Ω119871

119904(119878119899minus1)

sdot

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus120572119899(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840


le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus120572119899(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

119901

sdot 119908 (119911)119901119889119911)

1119901

sdot (int2119895+1119876

119908 (119911)minus1199011199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

sdot

infin

sum

119895=1

1

119908119902 (2119895+1119876)1119902minus120581119901

(71)


1198681198682le 119862

infin

sum

119895=1

int2119895+11198762119895119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

sdot1003816100381610038161003816Ω (119910 minus 119911)

1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572119889119911

+ 119862

infin

sum

119895=1

int2119895+11198762119895119876

sum

|120574|=119898minus1

100381610038161003816100381610038161198982119895+1119876(119863120574119860) minus 119898

119876(119863120574119860)10038161003816100381610038161003816

sdot1003816100381610038161003816Ω (119910 minus 119911)

1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572119889119911 fl 119868119868

21+ 119868119868

22

(72)


11986811986821le 119862

infin

sum

119895=1

1

100381610038161003816100381621198951198761003816100381610038161003816

1minus

120572

119899

int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863

120574119860)10038161003816100381610038161003816100381610038161003816Ω (119910 minus 119911)

10038161003816100381610038161003816100381610038161003816119891 (119911)

1003816100381610038161003816 119889119911

le 119862

infin

sum

119895=1

1

100381610038161003816100381621198951198761003816100381610038161003816

1minus

120572

119899

(int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

1199041015840

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840

(int2119895+1119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

119904119889119911)

1119904

le 119862 Ω119871119904(119878119899minus1)

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus

120572

119899

(int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

1199041015840

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840

le 119862 Ω119871119904(119878119899minus1)

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus

120572

119899

(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

119901119908 (119911)

119901119889119911)

1119901

sdot (int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

1199011199041015840

(119901minus1199041015840

)119908 (119911)

minus1199011199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

(73)


(int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

1199011199041015840

(119901minus1199041015840

)119908 (119911)

minus1199011199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

le 119862(int2119895+1119876

sum

|120574|=119898minus1

10038161003816100381610038161003816119863120574119860 (119911) minus 119898

2119895+1119876119908minus1199011199041015840(119901minus1199041015840)(119863120574119860)10038161003816100381610038161003816

1199011199041015840

(119901minus1199041015840

)

119908 (119911)minus1199011199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

+ sum

|120574|=119898minus1


2119895+1119876(119863120574119860)10038161003816100381610038161003816119908minus1199011199041015840

(119901minus1199041015840

)(2119895+1119876)(119901minus1199041015840

)1199011199041015840

fl 119868119868119868 + 119868119881

(74)


isin 119860(1199011199041015840 119902119904


119908minus1199011199041015840

(119901minus1199041015840

)isin 119860

1199051015840 sub 119860



(119901minus1199041015840


of BMO(R119899) and 1199081199041015840

isin 119860(1199011199041015840 119902119904

1015840) we have


119868119868119868 le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast 119908

minus1199011199041015840

(119901minus1199041015840

)(2119895+1119876)(119901minus1199041015840

)1199011199041015840

= 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

100381610038161003816100381610038162119895+111987610038161003816100381610038161003816

(119901minus1199041015840

)1199011199041015840

+1119902

119908119902 (2119895+1119876)1119902

(75)



119905 isin 2119895+1119876 sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816 gt 119904

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le 1198621

100381610038161003816100381610038162119895+111987610038161003816100381610038161003816119890minus1198622119904(sum|120574|=119898minus1

119863120574

119860lowast)

(76)



119908(

119905 isin 2119895+1119876 sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

gt 119904

) le 119862119908(2119895+1119876) 119890

minus1198622119904120575(sum

|120574|=119898minus1119863120574

119860lowast)

(77)


sum

|120574|=119898minus1


2119895+1119876(119863120574119860)10038161003816100381610038161003816

le1

119908minus1199011199041015840(119901minus119904

1015840) (2119895+1119876)

int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119905)

minus 1198982119895+1119876(119863120574119860)1003816100381610038161003816 119908minus1199011199041015840

(119901minus1199041015840

)(119905) 119889119905

=119862

119908minus1199011199041015840(119901minus119904

1015840) (2119895+1119876)

int

infin

0

119908minus1199011199041015840

(119901minus1199041015840

)(

119905

isin 2119895+1119876 sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

gt 119904

)119889119904 le119862

119908minus1199011199041015840(119901minus119904

1015840) (2119895+1119876)

sdot int

infin

0

119908minus1199011199041015840

(119901minus1199041015840

)(2119895+1119876) 119890

minus1198622119904120575(sum

|120574|=119898minus1119863120574

119860lowast)119889119904

= 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

(78)

As a result

119868119881 le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast 119908

minus1199011199041015840

(119901minus1199041015840

)(2119895+1119876)(119901minus1199041015840

)1199011199041015840

= 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

100381610038161003816100381610038162119895+111987610038161003816100381610038161003816

(119901minus1199041015840

)1199011199041015840

+1119902

119908119902 (2119895+1119876)1119902

(79)

Thus

11986811986821le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

sdot

infin

sum

119895=1

1

119908119902 (2119895+1119876)1119902minus120581119901

(80)


1199081199041015840

isin 119860(1199011199041015840 119902119904

1015840) we get

11986811986822le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

sdot

infin

sum

119895=1

119895 int2119895+11198762119895119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572119889119911

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast Ω119871

119904(119878119899minus1)

infin

sum

119895=1

119895

sdot

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus120572119899(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

infin

sum

119895=1

119895

sdot

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus120572119899(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

119901119908 (119911)

119901119889119911)

1119901

sdot (int2119895+1119876

119908 (119911)minus1199011199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

sdot

infin

sum

119895=1

119895

119908119902 (2119895+1119876)1119902minus120581119901

(81)

Hence

10038161003816100381610038161003816119879119860

Ω1205721198912(119910)10038161003816100381610038161003816le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

sdot

infin

sum

119895=1

119895

119908119902 (2119895+1119876)1119902minus120581119901

(82)

Therefore

119868119868 le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

sdot

infin

sum

119895=1

119895119908119902(119876)

1119902minus120581119901

119908119902 (2119895+1119876)1119902minus120581119901


le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

infin

sum

119895=1

119895

(119863119895+1

1)1119902minus120581119901

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

(83)


1

119908119902 (119876)120581119901(int119876

10038161003816100381610038161003816119879119860

Ω120572119891 (119910)

10038161003816100381610038161003816

119902

119908 (119910)119902119889119910)

1119902

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

(84)




119876+ 119891120594

(119876)119888 fl 119891

1+ 119891

2 Then we have

1

119908 (119876)120581119901(int119876

1003816100381610038161003816[119860 119879Ω] 119891 (119910)1003816100381610038161003816

119901119908 (119910) 119889119910)

1119901

le1

119908 (119876)120581119901(int119876

1003816100381610038161003816[119860 119879Ω] 1198911 (119910)1003816100381610038161003816

119901119908 (119910) 119889119910)

1119901

+1

119908 (119876)120581119901(int119876

1003816100381610038161003816[119860 119879Ω] 1198912 (119910)1003816100381610038161003816

119901119908 (119910) 119889119910)

1119901

fl 119868 + 119868119868

(85)


119868 le1

119908 (119876)120581119901

1003817100381710038171003817[119860 119879Ω] 11989111003817100381710038171003817119871119901(119908)

le119862

119908 (119876)120581119901119860lowast

100381710038171003817100381711989111003817100381710038171003817119871119901(119908)

= 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

119908(119876)120581119901

119908 (119876)120581119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(86)


1003816100381610038161003816[119860 119879Ω] 1198912 (119910)1003816100381610038161003816 le

infin

sum

119895=1

int2119895+11198762119895119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899

1003816100381610038161003816119860 (119910) minus 119860 (119911)1003816100381610038161003816

sdot1003816100381610038161003816119891 (119911)

1003816100381610038161003816 119889119911

le 119862

infin

sum

119895=1

1

100381610038161003816100381621198951198761003816100381610038161003816

(int2119895+1119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

119904119889119911)

1119904

sdot (int2119895+1119876

1003816100381610038161003816119860 (119910) minus 119860 (119911)1003816100381610038161003816

1199041015840

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840

le 119862 Ω119871119904

sdot

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

sdot (int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840

+ 119862 Ω119871119904

sdot

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

(int2119895+1119876


10038161003816100381610038161003816

1199041015840

sdot1003816100381610038161003816119891 (119911)

1003816100381610038161003816

1199041015840

119889119911)

11199041015840

fl 1198681198681(119910) + 119868119868

2

(87)




1

119908 (119876)120581119901(int119876

1198681198681(119910)

119901119908 (119910) 119889119910)

1119901

= 119862Ω119871119904

119908 (119876)120581119901

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

(int119876

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

119901

(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

1199011199041015840

119908 (119910) 119889119910)

1119901

le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

119908 (119876)

120581

119901

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

119908 (2119895+1119876)120581119901

119908minus1199041015840

(119901minus1199041015840

)(2119895+1119876)(119901minus1199041015840

)1199011199041015840

sdot (int119876

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

119908 (119876)120581119901

infin

sum

119895=1

1

119908 (2119895+1119876)(1minus120581)119901

(int119876

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

(88)



(int119876

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

le (int119876

1003816100381610038161003816119860 (119910) minus 119898119876119908 (119860)1003816100381610038161003816

119901119908 (119910) 119889119910)

1119901

+10038161003816100381610038161003816119898119876119908 (119860) minus 1198982119895+1119876119908minus119904

1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816119908 (119876)

1119901

fl 119868119868119868 + 119868119881

(89)



Lemma 25 that

119868119868119868 le 119860lowast 119908 (119876)1119901 (90)


10038161003816100381610038161003816119898119876119908 (119860) minus 1198982119895+1119876119908minus119904

1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

le1003816100381610038161003816119898119876119908 (119860) minus 119898119876 (119860)

1003816100381610038161003816 +1003816100381610038161003816119898119876 (119860) minus 1198982119895+1119876 (119860)

1003816100381610038161003816

+100381610038161003816100381610038161198982119895+1119876 (119860) minus 1198982119895+1119876119908minus119904

1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816le

1

119908 (119876)

sdot int119876

1003816100381610038161003816119860 (119905) minus 119898119876 (119860)1003816100381610038161003816 119908 (119905) 119889119905 + 2

119899(119895 + 1) 119860lowast

+1

119908minus1199041015840(119901minus119904

1015840) (2119895+1119876)

sdot int2119895+1119876

1003816100381610038161003816119860 (119905) minus 1198982119895+1119876 (119860)1003816100381610038161003816 119908minus1199041015840

(119901minus1199041015840

)(119905) 119889119905

le 119862 (119895 + 1) 119860lowast

(91)

Hence

119868119881 le 119862 (119895 + 1) 119860lowast 119908 (119876)1119901 (92)

As a result

1

119908 (119876)120581119901(int119876

1198681198681(119910)

119901119908 (119910) 119889119910)

1119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

infin

sum

119895=1

(119895 + 1)119908 (119876)

(1minus120581)119901

119908 (2119895+1119876)(1minus120581)119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(93)


1199011199041015840 we get

1198681198682le 119862 Ω119871119904

sdot

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

119901119908 (119911) 119889119911)

1119901

sdot (int2119895+1119876


10038161003816100381610038161003816

1199011199041015840

(119901minus1199041015840

)

sdot 119908 (119911)minus1199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

le 119862 Ω119871119904 119860lowast

sdot10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

119908 (2119895+1119876)120581119901

sdot 119908minus1199041015840

(119901minus1199041015840

)(2119895+1119876)(119901minus1199041015840

)1199011199041015840

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

sdot

infin

sum

119895=1

1

119908 (2119895+1119876)(1minus120581)119901

(94)

Therefore

1

119908 (119876)120581119901(int119876

119868119868119901

2119908 (119910) 119889119910)

1119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

infin

sum

119895=1

119908 (119876)(1minus120581)119901

119908 (2119895+1119876)(1minus120581)119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(95)




119860119876(119910) = 119860 (119910) minus 119898

119876(nabla119860) 119910 (96)


(119876)119888 fl

1198911+ 119891

2 Then we get

1

119908 (119876)120581119901(int119876

100381610038161003816100381610038161003816119860

Ω119891 (119910)

100381610038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

le1

119908 (119876)120581119901(int119876

100381610038161003816100381610038161003816119860

Ω1198911(119910)100381610038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

+1

119908 (119876)120581119901(int119876

100381610038161003816100381610038161003816119860

Ω1198912(119910)100381610038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

fl 119868 + 119868119868

(97)



119868 le1

119908 (119876)120581119901

100381710038171003817100381710038171003817119860

Ω1198911

100381710038171003817100381710038171003817119871119901(119908)

le119862


100381710038171003817100381711989111003817100381710038171003817119871119901(119908)

le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

119908(119876)120581119901

119908 (119876)120581119901

le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(98)


119901 Ω isin 119871




Competing Interests


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











10038161003816100381610038161003816119879Ω120572+120573

1198912 (119909)

10038161003816100381610038161003816=

infin

sum

119895=1

int2119895+11198612119895119861

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572minus120573

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

le 119862

infin

sum

119895=1

(int2119895+1119861

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

119904119889119910)

1119904

sdot (int2119895+11198612119895119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816

1199041015840

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

(119899minus120572minus120573)1199041015840119889119910)

11199041015840

fl 119862infin

sum

119895=1

(11986811198951198682119895)

(56)



1198681119895= (int

2119895+2119861

|Ω (119911)|119904119889119911)

1119904

= (int

2119895+2

119903119861

0

int119878119899minus1

10038161003816100381610038161003816Ω (119911

1015840)10038161003816100381610038161003816

119904

1198891199111015840119903119899minus1119889119903)

1119904

= 119862 Ω119871119904(119878119899minus1)

100381610038161003816100381610038162119895+211986110038161003816100381610038161003816

1119904

(57)


1198682119895le

119862

10038161003816100381610038162119895+1119861

1003816100381610038161003816

1minus(120572+120573)119899(int2119895+1119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816

1199041015840

119889119910)

11199041015840

(58)


isin 119860(1199011199041015840 119902119904

1015840) we get

(int2119895+1119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816

1199041015840

119889119910)

11199041015840

le 119862(int2119895+1119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119901119908 (119910)

119901119889119910)

1119901

sdot (int2119895+1119861

119908 (119910)minus1199011199041015840

(119901minus1199041015840

)119889119910)

(119901minus1199041015840

)1199011199041015840

le 11986210038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) 119908

119902(2119895+1119861)120581119901

sdot (int2119895+1119861

119908 (119910)minus1199011199041015840

(119901minus1199041015840

)119889119910)

(119901minus1199041015840

)1199011199041015840

le 11986210038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) 119908

119902(2119895+1119861)120581119901

sdot

100381610038161003816100381610038162119895+111986110038161003816100381610038161003816

(119901119902minus1199041015840

119902+1199041015840

119901)1199011199021199041015840

119908119902 (2119895+1119861)1119902

(59)

Thus

10038161003816100381610038161003816119879Ω120572+120573

1198912 (119909)

10038161003816100381610038161003816le 119862

infin

sum

119895=1

(11986811198951198682119895)

le 119862

infin

sum

119895=1

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

1

119908119902 (2119895+1119861)1119902minus120581119901

(60)

So we get

1198692le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

infin

sum

119895=1

119908119902(119861)

1119902minus120581119901

119908119902 (2119895+1119861)1119902minus120581119901

(61)


1198692le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) (62)



Ω120572is bounded




119879119860

Ω120572119891 (119909)

= intR119899

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572+119898minus1

1003816100381610038161003816119877119898 (119860 119909 119910)10038161003816100381610038161003816100381610038161003816119891 (119910)

1003816100381610038161003816 119889119910

0 le 120572 lt 119899

(63)


119860

Ω120572 On the


119879119860

Ω120572119891 (119909)

ge int|119909minus119910|lt119903

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572+119898minus1

1003816100381610038161003816119877119898 (119860 119909 119910)10038161003816100381610038161003816100381610038161003816119891 (119910)

1003816100381610038161003816 119889119910

ge1

119903119899minus120572+119898minus1


1003816100381610038161003816Ω (119909 minus 119910)10038161003816100381610038161003816100381610038161003816119877119898 (119860 119909 119910)

10038161003816100381610038161003816100381610038161003816119891 (119910)

1003816100381610038161003816 119889119910

(64)


119879119860

Ω120572119891 (119909) ge 119872

119860

Ω120572119891 (119909) (65)




119879Ω120573119891 (119909) = int

R119899

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120573

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910 (66)







119860119876(119910) = 119860 (119910) minus sum

|120574|=119898minus1

1

120574119898119876(119863120574119860)119910

120574 (67)


119876+ 119891120594

(119876)119888 = 119891

1+ 119891

2

Then we have

1

119908119902 (119876)120581119901(int119876

10038161003816100381610038161003816119879119860

Ω120572119891 (119910)

10038161003816100381610038161003816

119902

119908 (119910)119902119889119910)

1119902

le1

119908119902 (119876)120581119901(int119876

10038161003816100381610038161003816119879119860

Ω1205721198911(119910)10038161003816100381610038161003816

119902

119908 (119910)119902119889119910)

1119902

+1

119908119902 (119876)120581119901(int119876

10038161003816100381610038161003816119879119860

Ω1205721198912(119910)10038161003816100381610038161003816

119902

119908 (119910)119902119889119910)

1119902

fl 119868 + 119868119868

(68)


119868 le119862

119908119902 (119876)120581119901

sdot sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast (int

119876

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119901119908 (119910)

119901119889119910)

1119901

= 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)(

119908119902(119876)

119908119902 (119876))

120581119901

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

(69)


Ω1205721198912(119910) contained in 119868119868 By


10038161003816100381610038161003816119879119860

Ω1205721198912(119910)10038161003816100381610038161003816

le int(119876)119888

10038161003816100381610038161003816119877119898(119860119876 119910 119911)

100381610038161003816100381610038161003816100381610038161003816119910 minus 119911

1003816100381610038161003816

119898minus1

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

le 119862int(119876)119888

sum

|120574|=119898minus1

(1

10038161003816100381610038161003816119868119911119910

10038161003816100381610038161003816

int119868119911

119910

10038161003816100381610038161003816119863120574119860119876(119905)10038161003816100381610038161003816

119897

119889119905)

1119897

sdot1003816100381610038161003816Ω (119910 minus 119911)

1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572119889119911

+ 119862int(119876)119888

sum

|120574|=119898minus1

10038161003816100381610038161003816119863120574119860 (119911) minus 119898

119876(119863120574119860)10038161003816100381610038161003816

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

sdot

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572119889119911 fl 119868119868

1+ 119868119868

2

(70)

We estimate 1198681198681and 119868119868



isin 119860(1199011199041015840 119902119904

1015840) we get

1198681198681le 119862 sum

|120574|=119898minus1

(1

10038161003816100381610038161003816119868119911119910

10038161003816100381610038161003816

int119868119911

119910

10038161003816100381610038161003816119863120574119860 (119905) minus 119898

119876(119863120574119860)10038161003816100381610038161003816

119897

119889119905)

1119897

sdot

infin

sum

119895=1

int2119895+11198762119895119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572119889119911

le 119862 sum

|120574|=119898minus1

[

[

(1

10038161003816100381610038161003816119868119911119910

10038161003816100381610038161003816

sdot int119868119911

119910

100381610038161003816100381610038161003816119863120574119860 (119905) minus 119898119868119911

119910

(119863120574119860)100381610038161003816100381610038161003816

119897

119889119905)

1119897

+100381610038161003816100381610038161003816119898119868119911

119910

(119863120574119860)

minus 1198985119899119876(119863120574119860)100381610038161003816100381610038161003816+100381610038161003816100381610038161198985119899119876(119863120574119860) minus 119898

119876(119863120574119860)10038161003816100381610038161003816]

]

sdot

infin

sum

119895=1

(int2119895+1119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

119904119889119911)

1119904

sdot (int2119895+11198762119895119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

(119899minus120572)1199041015840119889119911)

11199041015840

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast Ω119871

119904(119878119899minus1)

sdot

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus120572119899(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840


le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus120572119899(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

119901

sdot 119908 (119911)119901119889119911)

1119901

sdot (int2119895+1119876

119908 (119911)minus1199011199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

sdot

infin

sum

119895=1

1

119908119902 (2119895+1119876)1119902minus120581119901

(71)


1198681198682le 119862

infin

sum

119895=1

int2119895+11198762119895119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

sdot1003816100381610038161003816Ω (119910 minus 119911)

1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572119889119911

+ 119862

infin

sum

119895=1

int2119895+11198762119895119876

sum

|120574|=119898minus1

100381610038161003816100381610038161198982119895+1119876(119863120574119860) minus 119898

119876(119863120574119860)10038161003816100381610038161003816

sdot1003816100381610038161003816Ω (119910 minus 119911)

1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572119889119911 fl 119868119868

21+ 119868119868

22

(72)


11986811986821le 119862

infin

sum

119895=1

1

100381610038161003816100381621198951198761003816100381610038161003816

1minus

120572

119899

int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863

120574119860)10038161003816100381610038161003816100381610038161003816Ω (119910 minus 119911)

10038161003816100381610038161003816100381610038161003816119891 (119911)

1003816100381610038161003816 119889119911

le 119862

infin

sum

119895=1

1

100381610038161003816100381621198951198761003816100381610038161003816

1minus

120572

119899

(int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

1199041015840

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840

(int2119895+1119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

119904119889119911)

1119904

le 119862 Ω119871119904(119878119899minus1)

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus

120572

119899

(int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

1199041015840

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840

le 119862 Ω119871119904(119878119899minus1)

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus

120572

119899

(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

119901119908 (119911)

119901119889119911)

1119901

sdot (int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

1199011199041015840

(119901minus1199041015840

)119908 (119911)

minus1199011199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

(73)


(int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

1199011199041015840

(119901minus1199041015840

)119908 (119911)

minus1199011199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

le 119862(int2119895+1119876

sum

|120574|=119898minus1

10038161003816100381610038161003816119863120574119860 (119911) minus 119898

2119895+1119876119908minus1199011199041015840(119901minus1199041015840)(119863120574119860)10038161003816100381610038161003816

1199011199041015840

(119901minus1199041015840

)

119908 (119911)minus1199011199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

+ sum

|120574|=119898minus1


2119895+1119876(119863120574119860)10038161003816100381610038161003816119908minus1199011199041015840

(119901minus1199041015840

)(2119895+1119876)(119901minus1199041015840

)1199011199041015840

fl 119868119868119868 + 119868119881

(74)


isin 119860(1199011199041015840 119902119904


119908minus1199011199041015840

(119901minus1199041015840

)isin 119860

1199051015840 sub 119860



(119901minus1199041015840


of BMO(R119899) and 1199081199041015840

isin 119860(1199011199041015840 119902119904

1015840) we have


119868119868119868 le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast 119908

minus1199011199041015840

(119901minus1199041015840

)(2119895+1119876)(119901minus1199041015840

)1199011199041015840

= 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

100381610038161003816100381610038162119895+111987610038161003816100381610038161003816

(119901minus1199041015840

)1199011199041015840

+1119902

119908119902 (2119895+1119876)1119902

(75)



119905 isin 2119895+1119876 sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816 gt 119904

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le 1198621

100381610038161003816100381610038162119895+111987610038161003816100381610038161003816119890minus1198622119904(sum|120574|=119898minus1

119863120574

119860lowast)

(76)



119908(

119905 isin 2119895+1119876 sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

gt 119904

) le 119862119908(2119895+1119876) 119890

minus1198622119904120575(sum

|120574|=119898minus1119863120574

119860lowast)

(77)


sum

|120574|=119898minus1


2119895+1119876(119863120574119860)10038161003816100381610038161003816

le1

119908minus1199011199041015840(119901minus119904

1015840) (2119895+1119876)

int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119905)

minus 1198982119895+1119876(119863120574119860)1003816100381610038161003816 119908minus1199011199041015840

(119901minus1199041015840

)(119905) 119889119905

=119862

119908minus1199011199041015840(119901minus119904

1015840) (2119895+1119876)

int

infin

0

119908minus1199011199041015840

(119901minus1199041015840

)(

119905

isin 2119895+1119876 sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

gt 119904

)119889119904 le119862

119908minus1199011199041015840(119901minus119904

1015840) (2119895+1119876)

sdot int

infin

0

119908minus1199011199041015840

(119901minus1199041015840

)(2119895+1119876) 119890

minus1198622119904120575(sum

|120574|=119898minus1119863120574

119860lowast)119889119904

= 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

(78)

As a result

119868119881 le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast 119908

minus1199011199041015840

(119901minus1199041015840

)(2119895+1119876)(119901minus1199041015840

)1199011199041015840

= 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

100381610038161003816100381610038162119895+111987610038161003816100381610038161003816

(119901minus1199041015840

)1199011199041015840

+1119902

119908119902 (2119895+1119876)1119902

(79)

Thus

11986811986821le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

sdot

infin

sum

119895=1

1

119908119902 (2119895+1119876)1119902minus120581119901

(80)


1199081199041015840

isin 119860(1199011199041015840 119902119904

1015840) we get

11986811986822le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

sdot

infin

sum

119895=1

119895 int2119895+11198762119895119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572119889119911

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast Ω119871

119904(119878119899minus1)

infin

sum

119895=1

119895

sdot

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus120572119899(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

infin

sum

119895=1

119895

sdot

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus120572119899(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

119901119908 (119911)

119901119889119911)

1119901

sdot (int2119895+1119876

119908 (119911)minus1199011199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

sdot

infin

sum

119895=1

119895

119908119902 (2119895+1119876)1119902minus120581119901

(81)

Hence

10038161003816100381610038161003816119879119860

Ω1205721198912(119910)10038161003816100381610038161003816le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

sdot

infin

sum

119895=1

119895

119908119902 (2119895+1119876)1119902minus120581119901

(82)

Therefore

119868119868 le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

sdot

infin

sum

119895=1

119895119908119902(119876)

1119902minus120581119901

119908119902 (2119895+1119876)1119902minus120581119901


le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

infin

sum

119895=1

119895

(119863119895+1

1)1119902minus120581119901

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

(83)


1

119908119902 (119876)120581119901(int119876

10038161003816100381610038161003816119879119860

Ω120572119891 (119910)

10038161003816100381610038161003816

119902

119908 (119910)119902119889119910)

1119902

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

(84)




119876+ 119891120594

(119876)119888 fl 119891

1+ 119891

2 Then we have

1

119908 (119876)120581119901(int119876

1003816100381610038161003816[119860 119879Ω] 119891 (119910)1003816100381610038161003816

119901119908 (119910) 119889119910)

1119901

le1

119908 (119876)120581119901(int119876

1003816100381610038161003816[119860 119879Ω] 1198911 (119910)1003816100381610038161003816

119901119908 (119910) 119889119910)

1119901

+1

119908 (119876)120581119901(int119876

1003816100381610038161003816[119860 119879Ω] 1198912 (119910)1003816100381610038161003816

119901119908 (119910) 119889119910)

1119901

fl 119868 + 119868119868

(85)


119868 le1

119908 (119876)120581119901

1003817100381710038171003817[119860 119879Ω] 11989111003817100381710038171003817119871119901(119908)

le119862

119908 (119876)120581119901119860lowast

100381710038171003817100381711989111003817100381710038171003817119871119901(119908)

= 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

119908(119876)120581119901

119908 (119876)120581119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(86)


1003816100381610038161003816[119860 119879Ω] 1198912 (119910)1003816100381610038161003816 le

infin

sum

119895=1

int2119895+11198762119895119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899

1003816100381610038161003816119860 (119910) minus 119860 (119911)1003816100381610038161003816

sdot1003816100381610038161003816119891 (119911)

1003816100381610038161003816 119889119911

le 119862

infin

sum

119895=1

1

100381610038161003816100381621198951198761003816100381610038161003816

(int2119895+1119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

119904119889119911)

1119904

sdot (int2119895+1119876

1003816100381610038161003816119860 (119910) minus 119860 (119911)1003816100381610038161003816

1199041015840

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840

le 119862 Ω119871119904

sdot

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

sdot (int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840

+ 119862 Ω119871119904

sdot

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

(int2119895+1119876


10038161003816100381610038161003816

1199041015840

sdot1003816100381610038161003816119891 (119911)

1003816100381610038161003816

1199041015840

119889119911)

11199041015840

fl 1198681198681(119910) + 119868119868

2

(87)




1

119908 (119876)120581119901(int119876

1198681198681(119910)

119901119908 (119910) 119889119910)

1119901

= 119862Ω119871119904

119908 (119876)120581119901

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

(int119876

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

119901

(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

1199011199041015840

119908 (119910) 119889119910)

1119901

le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

119908 (119876)

120581

119901

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

119908 (2119895+1119876)120581119901

119908minus1199041015840

(119901minus1199041015840

)(2119895+1119876)(119901minus1199041015840

)1199011199041015840

sdot (int119876

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

119908 (119876)120581119901

infin

sum

119895=1

1

119908 (2119895+1119876)(1minus120581)119901

(int119876

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

(88)



(int119876

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

le (int119876

1003816100381610038161003816119860 (119910) minus 119898119876119908 (119860)1003816100381610038161003816

119901119908 (119910) 119889119910)

1119901

+10038161003816100381610038161003816119898119876119908 (119860) minus 1198982119895+1119876119908minus119904

1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816119908 (119876)

1119901

fl 119868119868119868 + 119868119881

(89)



Lemma 25 that

119868119868119868 le 119860lowast 119908 (119876)1119901 (90)


10038161003816100381610038161003816119898119876119908 (119860) minus 1198982119895+1119876119908minus119904

1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

le1003816100381610038161003816119898119876119908 (119860) minus 119898119876 (119860)

1003816100381610038161003816 +1003816100381610038161003816119898119876 (119860) minus 1198982119895+1119876 (119860)

1003816100381610038161003816

+100381610038161003816100381610038161198982119895+1119876 (119860) minus 1198982119895+1119876119908minus119904

1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816le

1

119908 (119876)

sdot int119876

1003816100381610038161003816119860 (119905) minus 119898119876 (119860)1003816100381610038161003816 119908 (119905) 119889119905 + 2

119899(119895 + 1) 119860lowast

+1

119908minus1199041015840(119901minus119904

1015840) (2119895+1119876)

sdot int2119895+1119876

1003816100381610038161003816119860 (119905) minus 1198982119895+1119876 (119860)1003816100381610038161003816 119908minus1199041015840

(119901minus1199041015840

)(119905) 119889119905

le 119862 (119895 + 1) 119860lowast

(91)

Hence

119868119881 le 119862 (119895 + 1) 119860lowast 119908 (119876)1119901 (92)

As a result

1

119908 (119876)120581119901(int119876

1198681198681(119910)

119901119908 (119910) 119889119910)

1119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

infin

sum

119895=1

(119895 + 1)119908 (119876)

(1minus120581)119901

119908 (2119895+1119876)(1minus120581)119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(93)


1199011199041015840 we get

1198681198682le 119862 Ω119871119904

sdot

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

119901119908 (119911) 119889119911)

1119901

sdot (int2119895+1119876


10038161003816100381610038161003816

1199011199041015840

(119901minus1199041015840

)

sdot 119908 (119911)minus1199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

le 119862 Ω119871119904 119860lowast

sdot10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

119908 (2119895+1119876)120581119901

sdot 119908minus1199041015840

(119901minus1199041015840

)(2119895+1119876)(119901minus1199041015840

)1199011199041015840

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

sdot

infin

sum

119895=1

1

119908 (2119895+1119876)(1minus120581)119901

(94)

Therefore

1

119908 (119876)120581119901(int119876

119868119868119901

2119908 (119910) 119889119910)

1119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

infin

sum

119895=1

119908 (119876)(1minus120581)119901

119908 (2119895+1119876)(1minus120581)119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(95)




119860119876(119910) = 119860 (119910) minus 119898

119876(nabla119860) 119910 (96)


(119876)119888 fl

1198911+ 119891

2 Then we get

1

119908 (119876)120581119901(int119876

100381610038161003816100381610038161003816119860

Ω119891 (119910)

100381610038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

le1

119908 (119876)120581119901(int119876

100381610038161003816100381610038161003816119860

Ω1198911(119910)100381610038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

+1

119908 (119876)120581119901(int119876

100381610038161003816100381610038161003816119860

Ω1198912(119910)100381610038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

fl 119868 + 119868119868

(97)



119868 le1

119908 (119876)120581119901

100381710038171003817100381710038171003817119860

Ω1198911

100381710038171003817100381710038171003817119871119901(119908)

le119862


100381710038171003817100381711989111003817100381710038171003817119871119901(119908)

le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

119908(119876)120581119901

119908 (119876)120581119901

le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(98)


119901 Ω isin 119871




Competing Interests


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119879Ω120573119891 (119909) = int

R119899

1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120573

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910 (66)







119860119876(119910) = 119860 (119910) minus sum

|120574|=119898minus1

1

120574119898119876(119863120574119860)119910

120574 (67)


119876+ 119891120594

(119876)119888 = 119891

1+ 119891

2

Then we have

1

119908119902 (119876)120581119901(int119876

10038161003816100381610038161003816119879119860

Ω120572119891 (119910)

10038161003816100381610038161003816

119902

119908 (119910)119902119889119910)

1119902

le1

119908119902 (119876)120581119901(int119876

10038161003816100381610038161003816119879119860

Ω1205721198911(119910)10038161003816100381610038161003816

119902

119908 (119910)119902119889119910)

1119902

+1

119908119902 (119876)120581119901(int119876

10038161003816100381610038161003816119879119860

Ω1205721198912(119910)10038161003816100381610038161003816

119902

119908 (119910)119902119889119910)

1119902

fl 119868 + 119868119868

(68)


119868 le119862

119908119902 (119876)120581119901

sdot sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast (int

119876

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119901119908 (119910)

119901119889119910)

1119901

= 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)(

119908119902(119876)

119908119902 (119876))

120581119901

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

(69)


Ω1205721198912(119910) contained in 119868119868 By


10038161003816100381610038161003816119879119860

Ω1205721198912(119910)10038161003816100381610038161003816

le int(119876)119888

10038161003816100381610038161003816119877119898(119860119876 119910 119911)

100381610038161003816100381610038161003816100381610038161003816119910 minus 119911

1003816100381610038161003816

119898minus1

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

le 119862int(119876)119888

sum

|120574|=119898minus1

(1

10038161003816100381610038161003816119868119911119910

10038161003816100381610038161003816

int119868119911

119910

10038161003816100381610038161003816119863120574119860119876(119905)10038161003816100381610038161003816

119897

119889119905)

1119897

sdot1003816100381610038161003816Ω (119910 minus 119911)

1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572119889119911

+ 119862int(119876)119888

sum

|120574|=119898minus1

10038161003816100381610038161003816119863120574119860 (119911) minus 119898

119876(119863120574119860)10038161003816100381610038161003816

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

sdot

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572119889119911 fl 119868119868

1+ 119868119868

2

(70)

We estimate 1198681198681and 119868119868



isin 119860(1199011199041015840 119902119904

1015840) we get

1198681198681le 119862 sum

|120574|=119898minus1

(1

10038161003816100381610038161003816119868119911119910

10038161003816100381610038161003816

int119868119911

119910

10038161003816100381610038161003816119863120574119860 (119905) minus 119898

119876(119863120574119860)10038161003816100381610038161003816

119897

119889119905)

1119897

sdot

infin

sum

119895=1

int2119895+11198762119895119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572119889119911

le 119862 sum

|120574|=119898minus1

[

[

(1

10038161003816100381610038161003816119868119911119910

10038161003816100381610038161003816

sdot int119868119911

119910

100381610038161003816100381610038161003816119863120574119860 (119905) minus 119898119868119911

119910

(119863120574119860)100381610038161003816100381610038161003816

119897

119889119905)

1119897

+100381610038161003816100381610038161003816119898119868119911

119910

(119863120574119860)

minus 1198985119899119876(119863120574119860)100381610038161003816100381610038161003816+100381610038161003816100381610038161198985119899119876(119863120574119860) minus 119898

119876(119863120574119860)10038161003816100381610038161003816]

]

sdot

infin

sum

119895=1

(int2119895+1119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

119904119889119911)

1119904

sdot (int2119895+11198762119895119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

(119899minus120572)1199041015840119889119911)

11199041015840

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast Ω119871

119904(119878119899minus1)

sdot

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus120572119899(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840


le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus120572119899(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

119901

sdot 119908 (119911)119901119889119911)

1119901

sdot (int2119895+1119876

119908 (119911)minus1199011199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

sdot

infin

sum

119895=1

1

119908119902 (2119895+1119876)1119902minus120581119901

(71)


1198681198682le 119862

infin

sum

119895=1

int2119895+11198762119895119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

sdot1003816100381610038161003816Ω (119910 minus 119911)

1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572119889119911

+ 119862

infin

sum

119895=1

int2119895+11198762119895119876

sum

|120574|=119898minus1

100381610038161003816100381610038161198982119895+1119876(119863120574119860) minus 119898

119876(119863120574119860)10038161003816100381610038161003816

sdot1003816100381610038161003816Ω (119910 minus 119911)

1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572119889119911 fl 119868119868

21+ 119868119868

22

(72)


11986811986821le 119862

infin

sum

119895=1

1

100381610038161003816100381621198951198761003816100381610038161003816

1minus

120572

119899

int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863

120574119860)10038161003816100381610038161003816100381610038161003816Ω (119910 minus 119911)

10038161003816100381610038161003816100381610038161003816119891 (119911)

1003816100381610038161003816 119889119911

le 119862

infin

sum

119895=1

1

100381610038161003816100381621198951198761003816100381610038161003816

1minus

120572

119899

(int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

1199041015840

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840

(int2119895+1119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

119904119889119911)

1119904

le 119862 Ω119871119904(119878119899minus1)

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus

120572

119899

(int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

1199041015840

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840

le 119862 Ω119871119904(119878119899minus1)

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus

120572

119899

(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

119901119908 (119911)

119901119889119911)

1119901

sdot (int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

1199011199041015840

(119901minus1199041015840

)119908 (119911)

minus1199011199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

(73)


(int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

1199011199041015840

(119901minus1199041015840

)119908 (119911)

minus1199011199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

le 119862(int2119895+1119876

sum

|120574|=119898minus1

10038161003816100381610038161003816119863120574119860 (119911) minus 119898

2119895+1119876119908minus1199011199041015840(119901minus1199041015840)(119863120574119860)10038161003816100381610038161003816

1199011199041015840

(119901minus1199041015840

)

119908 (119911)minus1199011199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

+ sum

|120574|=119898minus1


2119895+1119876(119863120574119860)10038161003816100381610038161003816119908minus1199011199041015840

(119901minus1199041015840

)(2119895+1119876)(119901minus1199041015840

)1199011199041015840

fl 119868119868119868 + 119868119881

(74)


isin 119860(1199011199041015840 119902119904


119908minus1199011199041015840

(119901minus1199041015840

)isin 119860

1199051015840 sub 119860



(119901minus1199041015840


of BMO(R119899) and 1199081199041015840

isin 119860(1199011199041015840 119902119904

1015840) we have


119868119868119868 le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast 119908

minus1199011199041015840

(119901minus1199041015840

)(2119895+1119876)(119901minus1199041015840

)1199011199041015840

= 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

100381610038161003816100381610038162119895+111987610038161003816100381610038161003816

(119901minus1199041015840

)1199011199041015840

+1119902

119908119902 (2119895+1119876)1119902

(75)



119905 isin 2119895+1119876 sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816 gt 119904

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le 1198621

100381610038161003816100381610038162119895+111987610038161003816100381610038161003816119890minus1198622119904(sum|120574|=119898minus1

119863120574

119860lowast)

(76)



119908(

119905 isin 2119895+1119876 sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

gt 119904

) le 119862119908(2119895+1119876) 119890

minus1198622119904120575(sum

|120574|=119898minus1119863120574

119860lowast)

(77)


sum

|120574|=119898minus1


2119895+1119876(119863120574119860)10038161003816100381610038161003816

le1

119908minus1199011199041015840(119901minus119904

1015840) (2119895+1119876)

int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119905)

minus 1198982119895+1119876(119863120574119860)1003816100381610038161003816 119908minus1199011199041015840

(119901minus1199041015840

)(119905) 119889119905

=119862

119908minus1199011199041015840(119901minus119904

1015840) (2119895+1119876)

int

infin

0

119908minus1199011199041015840

(119901minus1199041015840

)(

119905

isin 2119895+1119876 sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

gt 119904

)119889119904 le119862

119908minus1199011199041015840(119901minus119904

1015840) (2119895+1119876)

sdot int

infin

0

119908minus1199011199041015840

(119901minus1199041015840

)(2119895+1119876) 119890

minus1198622119904120575(sum

|120574|=119898minus1119863120574

119860lowast)119889119904

= 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

(78)

As a result

119868119881 le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast 119908

minus1199011199041015840

(119901minus1199041015840

)(2119895+1119876)(119901minus1199041015840

)1199011199041015840

= 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

100381610038161003816100381610038162119895+111987610038161003816100381610038161003816

(119901minus1199041015840

)1199011199041015840

+1119902

119908119902 (2119895+1119876)1119902

(79)

Thus

11986811986821le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

sdot

infin

sum

119895=1

1

119908119902 (2119895+1119876)1119902minus120581119901

(80)


1199081199041015840

isin 119860(1199011199041015840 119902119904

1015840) we get

11986811986822le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

sdot

infin

sum

119895=1

119895 int2119895+11198762119895119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572119889119911

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast Ω119871

119904(119878119899minus1)

infin

sum

119895=1

119895

sdot

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus120572119899(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

infin

sum

119895=1

119895

sdot

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus120572119899(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

119901119908 (119911)

119901119889119911)

1119901

sdot (int2119895+1119876

119908 (119911)minus1199011199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

sdot

infin

sum

119895=1

119895

119908119902 (2119895+1119876)1119902minus120581119901

(81)

Hence

10038161003816100381610038161003816119879119860

Ω1205721198912(119910)10038161003816100381610038161003816le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

sdot

infin

sum

119895=1

119895

119908119902 (2119895+1119876)1119902minus120581119901

(82)

Therefore

119868119868 le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

sdot

infin

sum

119895=1

119895119908119902(119876)

1119902minus120581119901

119908119902 (2119895+1119876)1119902minus120581119901


le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

infin

sum

119895=1

119895

(119863119895+1

1)1119902minus120581119901

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

(83)


1

119908119902 (119876)120581119901(int119876

10038161003816100381610038161003816119879119860

Ω120572119891 (119910)

10038161003816100381610038161003816

119902

119908 (119910)119902119889119910)

1119902

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

(84)




119876+ 119891120594

(119876)119888 fl 119891

1+ 119891

2 Then we have

1

119908 (119876)120581119901(int119876

1003816100381610038161003816[119860 119879Ω] 119891 (119910)1003816100381610038161003816

119901119908 (119910) 119889119910)

1119901

le1

119908 (119876)120581119901(int119876

1003816100381610038161003816[119860 119879Ω] 1198911 (119910)1003816100381610038161003816

119901119908 (119910) 119889119910)

1119901

+1

119908 (119876)120581119901(int119876

1003816100381610038161003816[119860 119879Ω] 1198912 (119910)1003816100381610038161003816

119901119908 (119910) 119889119910)

1119901

fl 119868 + 119868119868

(85)


119868 le1

119908 (119876)120581119901

1003817100381710038171003817[119860 119879Ω] 11989111003817100381710038171003817119871119901(119908)

le119862

119908 (119876)120581119901119860lowast

100381710038171003817100381711989111003817100381710038171003817119871119901(119908)

= 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

119908(119876)120581119901

119908 (119876)120581119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(86)


1003816100381610038161003816[119860 119879Ω] 1198912 (119910)1003816100381610038161003816 le

infin

sum

119895=1

int2119895+11198762119895119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899

1003816100381610038161003816119860 (119910) minus 119860 (119911)1003816100381610038161003816

sdot1003816100381610038161003816119891 (119911)

1003816100381610038161003816 119889119911

le 119862

infin

sum

119895=1

1

100381610038161003816100381621198951198761003816100381610038161003816

(int2119895+1119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

119904119889119911)

1119904

sdot (int2119895+1119876

1003816100381610038161003816119860 (119910) minus 119860 (119911)1003816100381610038161003816

1199041015840

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840

le 119862 Ω119871119904

sdot

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

sdot (int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840

+ 119862 Ω119871119904

sdot

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

(int2119895+1119876


10038161003816100381610038161003816

1199041015840

sdot1003816100381610038161003816119891 (119911)

1003816100381610038161003816

1199041015840

119889119911)

11199041015840

fl 1198681198681(119910) + 119868119868

2

(87)




1

119908 (119876)120581119901(int119876

1198681198681(119910)

119901119908 (119910) 119889119910)

1119901

= 119862Ω119871119904

119908 (119876)120581119901

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

(int119876

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

119901

(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

1199011199041015840

119908 (119910) 119889119910)

1119901

le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

119908 (119876)

120581

119901

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

119908 (2119895+1119876)120581119901

119908minus1199041015840

(119901minus1199041015840

)(2119895+1119876)(119901minus1199041015840

)1199011199041015840

sdot (int119876

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

119908 (119876)120581119901

infin

sum

119895=1

1

119908 (2119895+1119876)(1minus120581)119901

(int119876

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

(88)



(int119876

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

le (int119876

1003816100381610038161003816119860 (119910) minus 119898119876119908 (119860)1003816100381610038161003816

119901119908 (119910) 119889119910)

1119901

+10038161003816100381610038161003816119898119876119908 (119860) minus 1198982119895+1119876119908minus119904

1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816119908 (119876)

1119901

fl 119868119868119868 + 119868119881

(89)



Lemma 25 that

119868119868119868 le 119860lowast 119908 (119876)1119901 (90)


10038161003816100381610038161003816119898119876119908 (119860) minus 1198982119895+1119876119908minus119904

1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

le1003816100381610038161003816119898119876119908 (119860) minus 119898119876 (119860)

1003816100381610038161003816 +1003816100381610038161003816119898119876 (119860) minus 1198982119895+1119876 (119860)

1003816100381610038161003816

+100381610038161003816100381610038161198982119895+1119876 (119860) minus 1198982119895+1119876119908minus119904

1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816le

1

119908 (119876)

sdot int119876

1003816100381610038161003816119860 (119905) minus 119898119876 (119860)1003816100381610038161003816 119908 (119905) 119889119905 + 2

119899(119895 + 1) 119860lowast

+1

119908minus1199041015840(119901minus119904

1015840) (2119895+1119876)

sdot int2119895+1119876

1003816100381610038161003816119860 (119905) minus 1198982119895+1119876 (119860)1003816100381610038161003816 119908minus1199041015840

(119901minus1199041015840

)(119905) 119889119905

le 119862 (119895 + 1) 119860lowast

(91)

Hence

119868119881 le 119862 (119895 + 1) 119860lowast 119908 (119876)1119901 (92)

As a result

1

119908 (119876)120581119901(int119876

1198681198681(119910)

119901119908 (119910) 119889119910)

1119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

infin

sum

119895=1

(119895 + 1)119908 (119876)

(1minus120581)119901

119908 (2119895+1119876)(1minus120581)119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(93)


1199011199041015840 we get

1198681198682le 119862 Ω119871119904

sdot

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

119901119908 (119911) 119889119911)

1119901

sdot (int2119895+1119876


10038161003816100381610038161003816

1199011199041015840

(119901minus1199041015840

)

sdot 119908 (119911)minus1199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

le 119862 Ω119871119904 119860lowast

sdot10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

119908 (2119895+1119876)120581119901

sdot 119908minus1199041015840

(119901minus1199041015840

)(2119895+1119876)(119901minus1199041015840

)1199011199041015840

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

sdot

infin

sum

119895=1

1

119908 (2119895+1119876)(1minus120581)119901

(94)

Therefore

1

119908 (119876)120581119901(int119876

119868119868119901

2119908 (119910) 119889119910)

1119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

infin

sum

119895=1

119908 (119876)(1minus120581)119901

119908 (2119895+1119876)(1minus120581)119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(95)




119860119876(119910) = 119860 (119910) minus 119898

119876(nabla119860) 119910 (96)


(119876)119888 fl

1198911+ 119891

2 Then we get

1

119908 (119876)120581119901(int119876

100381610038161003816100381610038161003816119860

Ω119891 (119910)

100381610038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

le1

119908 (119876)120581119901(int119876

100381610038161003816100381610038161003816119860

Ω1198911(119910)100381610038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

+1

119908 (119876)120581119901(int119876

100381610038161003816100381610038161003816119860

Ω1198912(119910)100381610038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

fl 119868 + 119868119868

(97)



119868 le1

119908 (119876)120581119901

100381710038171003817100381710038171003817119860

Ω1198911

100381710038171003817100381710038171003817119871119901(119908)

le119862


100381710038171003817100381711989111003817100381710038171003817119871119901(119908)

le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

119908(119876)120581119901

119908 (119876)120581119901

le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(98)


119901 Ω isin 119871




Competing Interests


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus120572119899(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

119901

sdot 119908 (119911)119901119889119911)

1119901

sdot (int2119895+1119876

119908 (119911)minus1199011199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

sdot

infin

sum

119895=1

1

119908119902 (2119895+1119876)1119902minus120581119901

(71)


1198681198682le 119862

infin

sum

119895=1

int2119895+11198762119895119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

sdot1003816100381610038161003816Ω (119910 minus 119911)

1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572119889119911

+ 119862

infin

sum

119895=1

int2119895+11198762119895119876

sum

|120574|=119898minus1

100381610038161003816100381610038161198982119895+1119876(119863120574119860) minus 119898

119876(119863120574119860)10038161003816100381610038161003816

sdot1003816100381610038161003816Ω (119910 minus 119911)

1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572119889119911 fl 119868119868

21+ 119868119868

22

(72)


11986811986821le 119862

infin

sum

119895=1

1

100381610038161003816100381621198951198761003816100381610038161003816

1minus

120572

119899

int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863

120574119860)10038161003816100381610038161003816100381610038161003816Ω (119910 minus 119911)

10038161003816100381610038161003816100381610038161003816119891 (119911)

1003816100381610038161003816 119889119911

le 119862

infin

sum

119895=1

1

100381610038161003816100381621198951198761003816100381610038161003816

1minus

120572

119899

(int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

1199041015840

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840

(int2119895+1119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

119904119889119911)

1119904

le 119862 Ω119871119904(119878119899minus1)

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus

120572

119899

(int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

1199041015840

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840

le 119862 Ω119871119904(119878119899minus1)

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus

120572

119899

(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

119901119908 (119911)

119901119889119911)

1119901

sdot (int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

1199011199041015840

(119901minus1199041015840

)119908 (119911)

minus1199011199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

(73)


(int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

1199011199041015840

(119901minus1199041015840

)119908 (119911)

minus1199011199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

le 119862(int2119895+1119876

sum

|120574|=119898minus1

10038161003816100381610038161003816119863120574119860 (119911) minus 119898

2119895+1119876119908minus1199011199041015840(119901minus1199041015840)(119863120574119860)10038161003816100381610038161003816

1199011199041015840

(119901minus1199041015840

)

119908 (119911)minus1199011199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

+ sum

|120574|=119898minus1


2119895+1119876(119863120574119860)10038161003816100381610038161003816119908minus1199011199041015840

(119901minus1199041015840

)(2119895+1119876)(119901minus1199041015840

)1199011199041015840

fl 119868119868119868 + 119868119881

(74)


isin 119860(1199011199041015840 119902119904


119908minus1199011199041015840

(119901minus1199041015840

)isin 119860

1199051015840 sub 119860



(119901minus1199041015840


of BMO(R119899) and 1199081199041015840

isin 119860(1199011199041015840 119902119904

1015840) we have


119868119868119868 le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast 119908

minus1199011199041015840

(119901minus1199041015840

)(2119895+1119876)(119901minus1199041015840

)1199011199041015840

= 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

100381610038161003816100381610038162119895+111987610038161003816100381610038161003816

(119901minus1199041015840

)1199011199041015840

+1119902

119908119902 (2119895+1119876)1119902

(75)



119905 isin 2119895+1119876 sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816 gt 119904

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le 1198621

100381610038161003816100381610038162119895+111987610038161003816100381610038161003816119890minus1198622119904(sum|120574|=119898minus1

119863120574

119860lowast)

(76)



119908(

119905 isin 2119895+1119876 sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

gt 119904

) le 119862119908(2119895+1119876) 119890

minus1198622119904120575(sum

|120574|=119898minus1119863120574

119860lowast)

(77)


sum

|120574|=119898minus1


2119895+1119876(119863120574119860)10038161003816100381610038161003816

le1

119908minus1199011199041015840(119901minus119904

1015840) (2119895+1119876)

int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119905)

minus 1198982119895+1119876(119863120574119860)1003816100381610038161003816 119908minus1199011199041015840

(119901minus1199041015840

)(119905) 119889119905

=119862

119908minus1199011199041015840(119901minus119904

1015840) (2119895+1119876)

int

infin

0

119908minus1199011199041015840

(119901minus1199041015840

)(

119905

isin 2119895+1119876 sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

gt 119904

)119889119904 le119862

119908minus1199011199041015840(119901minus119904

1015840) (2119895+1119876)

sdot int

infin

0

119908minus1199011199041015840

(119901minus1199041015840

)(2119895+1119876) 119890

minus1198622119904120575(sum

|120574|=119898minus1119863120574

119860lowast)119889119904

= 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

(78)

As a result

119868119881 le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast 119908

minus1199011199041015840

(119901minus1199041015840

)(2119895+1119876)(119901minus1199041015840

)1199011199041015840

= 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

100381610038161003816100381610038162119895+111987610038161003816100381610038161003816

(119901minus1199041015840

)1199011199041015840

+1119902

119908119902 (2119895+1119876)1119902

(79)

Thus

11986811986821le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

sdot

infin

sum

119895=1

1

119908119902 (2119895+1119876)1119902minus120581119901

(80)


1199081199041015840

isin 119860(1199011199041015840 119902119904

1015840) we get

11986811986822le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

sdot

infin

sum

119895=1

119895 int2119895+11198762119895119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572119889119911

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast Ω119871

119904(119878119899minus1)

infin

sum

119895=1

119895

sdot

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus120572119899(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

infin

sum

119895=1

119895

sdot

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus120572119899(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

119901119908 (119911)

119901119889119911)

1119901

sdot (int2119895+1119876

119908 (119911)minus1199011199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

sdot

infin

sum

119895=1

119895

119908119902 (2119895+1119876)1119902minus120581119901

(81)

Hence

10038161003816100381610038161003816119879119860

Ω1205721198912(119910)10038161003816100381610038161003816le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

sdot

infin

sum

119895=1

119895

119908119902 (2119895+1119876)1119902minus120581119901

(82)

Therefore

119868119868 le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

sdot

infin

sum

119895=1

119895119908119902(119876)

1119902minus120581119901

119908119902 (2119895+1119876)1119902minus120581119901


le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

infin

sum

119895=1

119895

(119863119895+1

1)1119902minus120581119901

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

(83)


1

119908119902 (119876)120581119901(int119876

10038161003816100381610038161003816119879119860

Ω120572119891 (119910)

10038161003816100381610038161003816

119902

119908 (119910)119902119889119910)

1119902

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

(84)




119876+ 119891120594

(119876)119888 fl 119891

1+ 119891

2 Then we have

1

119908 (119876)120581119901(int119876

1003816100381610038161003816[119860 119879Ω] 119891 (119910)1003816100381610038161003816

119901119908 (119910) 119889119910)

1119901

le1

119908 (119876)120581119901(int119876

1003816100381610038161003816[119860 119879Ω] 1198911 (119910)1003816100381610038161003816

119901119908 (119910) 119889119910)

1119901

+1

119908 (119876)120581119901(int119876

1003816100381610038161003816[119860 119879Ω] 1198912 (119910)1003816100381610038161003816

119901119908 (119910) 119889119910)

1119901

fl 119868 + 119868119868

(85)


119868 le1

119908 (119876)120581119901

1003817100381710038171003817[119860 119879Ω] 11989111003817100381710038171003817119871119901(119908)

le119862

119908 (119876)120581119901119860lowast

100381710038171003817100381711989111003817100381710038171003817119871119901(119908)

= 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

119908(119876)120581119901

119908 (119876)120581119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(86)


1003816100381610038161003816[119860 119879Ω] 1198912 (119910)1003816100381610038161003816 le

infin

sum

119895=1

int2119895+11198762119895119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899

1003816100381610038161003816119860 (119910) minus 119860 (119911)1003816100381610038161003816

sdot1003816100381610038161003816119891 (119911)

1003816100381610038161003816 119889119911

le 119862

infin

sum

119895=1

1

100381610038161003816100381621198951198761003816100381610038161003816

(int2119895+1119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

119904119889119911)

1119904

sdot (int2119895+1119876

1003816100381610038161003816119860 (119910) minus 119860 (119911)1003816100381610038161003816

1199041015840

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840

le 119862 Ω119871119904

sdot

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

sdot (int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840

+ 119862 Ω119871119904

sdot

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

(int2119895+1119876


10038161003816100381610038161003816

1199041015840

sdot1003816100381610038161003816119891 (119911)

1003816100381610038161003816

1199041015840

119889119911)

11199041015840

fl 1198681198681(119910) + 119868119868

2

(87)




1

119908 (119876)120581119901(int119876

1198681198681(119910)

119901119908 (119910) 119889119910)

1119901

= 119862Ω119871119904

119908 (119876)120581119901

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

(int119876

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

119901

(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

1199011199041015840

119908 (119910) 119889119910)

1119901

le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

119908 (119876)

120581

119901

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

119908 (2119895+1119876)120581119901

119908minus1199041015840

(119901minus1199041015840

)(2119895+1119876)(119901minus1199041015840

)1199011199041015840

sdot (int119876

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

119908 (119876)120581119901

infin

sum

119895=1

1

119908 (2119895+1119876)(1minus120581)119901

(int119876

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

(88)



(int119876

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

le (int119876

1003816100381610038161003816119860 (119910) minus 119898119876119908 (119860)1003816100381610038161003816

119901119908 (119910) 119889119910)

1119901

+10038161003816100381610038161003816119898119876119908 (119860) minus 1198982119895+1119876119908minus119904

1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816119908 (119876)

1119901

fl 119868119868119868 + 119868119881

(89)



Lemma 25 that

119868119868119868 le 119860lowast 119908 (119876)1119901 (90)


10038161003816100381610038161003816119898119876119908 (119860) minus 1198982119895+1119876119908minus119904

1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

le1003816100381610038161003816119898119876119908 (119860) minus 119898119876 (119860)

1003816100381610038161003816 +1003816100381610038161003816119898119876 (119860) minus 1198982119895+1119876 (119860)

1003816100381610038161003816

+100381610038161003816100381610038161198982119895+1119876 (119860) minus 1198982119895+1119876119908minus119904

1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816le

1

119908 (119876)

sdot int119876

1003816100381610038161003816119860 (119905) minus 119898119876 (119860)1003816100381610038161003816 119908 (119905) 119889119905 + 2

119899(119895 + 1) 119860lowast

+1

119908minus1199041015840(119901minus119904

1015840) (2119895+1119876)

sdot int2119895+1119876

1003816100381610038161003816119860 (119905) minus 1198982119895+1119876 (119860)1003816100381610038161003816 119908minus1199041015840

(119901minus1199041015840

)(119905) 119889119905

le 119862 (119895 + 1) 119860lowast

(91)

Hence

119868119881 le 119862 (119895 + 1) 119860lowast 119908 (119876)1119901 (92)

As a result

1

119908 (119876)120581119901(int119876

1198681198681(119910)

119901119908 (119910) 119889119910)

1119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

infin

sum

119895=1

(119895 + 1)119908 (119876)

(1minus120581)119901

119908 (2119895+1119876)(1minus120581)119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(93)


1199011199041015840 we get

1198681198682le 119862 Ω119871119904

sdot

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

119901119908 (119911) 119889119911)

1119901

sdot (int2119895+1119876


10038161003816100381610038161003816

1199011199041015840

(119901minus1199041015840

)

sdot 119908 (119911)minus1199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

le 119862 Ω119871119904 119860lowast

sdot10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

119908 (2119895+1119876)120581119901

sdot 119908minus1199041015840

(119901minus1199041015840

)(2119895+1119876)(119901minus1199041015840

)1199011199041015840

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

sdot

infin

sum

119895=1

1

119908 (2119895+1119876)(1minus120581)119901

(94)

Therefore

1

119908 (119876)120581119901(int119876

119868119868119901

2119908 (119910) 119889119910)

1119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

infin

sum

119895=1

119908 (119876)(1minus120581)119901

119908 (2119895+1119876)(1minus120581)119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(95)




119860119876(119910) = 119860 (119910) minus 119898

119876(nabla119860) 119910 (96)


(119876)119888 fl

1198911+ 119891

2 Then we get

1

119908 (119876)120581119901(int119876

100381610038161003816100381610038161003816119860

Ω119891 (119910)

100381610038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

le1

119908 (119876)120581119901(int119876

100381610038161003816100381610038161003816119860

Ω1198911(119910)100381610038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

+1

119908 (119876)120581119901(int119876

100381610038161003816100381610038161003816119860

Ω1198912(119910)100381610038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

fl 119868 + 119868119868

(97)



119868 le1

119908 (119876)120581119901

100381710038171003817100381710038171003817119860

Ω1198911

100381710038171003817100381710038171003817119871119901(119908)

le119862


100381710038171003817100381711989111003817100381710038171003817119871119901(119908)

le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

119908(119876)120581119901

119908 (119876)120581119901

le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(98)


119901 Ω isin 119871




Competing Interests


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119868119868119868 le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast 119908

minus1199011199041015840

(119901minus1199041015840

)(2119895+1119876)(119901minus1199041015840

)1199011199041015840

= 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

100381610038161003816100381610038162119895+111987610038161003816100381610038161003816

(119901minus1199041015840

)1199011199041015840

+1119902

119908119902 (2119895+1119876)1119902

(75)



119905 isin 2119895+1119876 sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816 gt 119904

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le 1198621

100381610038161003816100381610038162119895+111987610038161003816100381610038161003816119890minus1198622119904(sum|120574|=119898minus1

119863120574

119860lowast)

(76)



119908(

119905 isin 2119895+1119876 sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

gt 119904

) le 119862119908(2119895+1119876) 119890

minus1198622119904120575(sum

|120574|=119898minus1119863120574

119860lowast)

(77)


sum

|120574|=119898minus1


2119895+1119876(119863120574119860)10038161003816100381610038161003816

le1

119908minus1199011199041015840(119901minus119904

1015840) (2119895+1119876)

int2119895+1119876

sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119905)

minus 1198982119895+1119876(119863120574119860)1003816100381610038161003816 119908minus1199011199041015840

(119901minus1199041015840

)(119905) 119889119905

=119862

119908minus1199011199041015840(119901minus119904

1015840) (2119895+1119876)

int

infin

0

119908minus1199011199041015840

(119901minus1199041015840

)(

119905

isin 2119895+1119876 sum

|120574|=119898minus1

1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863

120574119860)1003816100381610038161003816

gt 119904

)119889119904 le119862

119908minus1199011199041015840(119901minus119904

1015840) (2119895+1119876)

sdot int

infin

0

119908minus1199011199041015840

(119901minus1199041015840

)(2119895+1119876) 119890

minus1198622119904120575(sum

|120574|=119898minus1119863120574

119860lowast)119889119904

= 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

(78)

As a result

119868119881 le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast 119908

minus1199011199041015840

(119901minus1199041015840

)(2119895+1119876)(119901minus1199041015840

)1199011199041015840

= 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

100381610038161003816100381610038162119895+111987610038161003816100381610038161003816

(119901minus1199041015840

)1199011199041015840

+1119902

119908119902 (2119895+1119876)1119902

(79)

Thus

11986811986821le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

sdot

infin

sum

119895=1

1

119908119902 (2119895+1119876)1119902minus120581119901

(80)


1199081199041015840

isin 119860(1199011199041015840 119902119904

1015840) we get

11986811986822le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

sdot

infin

sum

119895=1

119895 int2119895+11198762119895119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899minus120572119889119911

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast Ω119871

119904(119878119899minus1)

infin

sum

119895=1

119895

sdot

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus120572119899(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

infin

sum

119895=1

119895

sdot

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

1minus120572119899(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

119901119908 (119911)

119901119889119911)

1119901

sdot (int2119895+1119876

119908 (119911)minus1199011199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

sdot

infin

sum

119895=1

119895

119908119902 (2119895+1119876)1119902minus120581119901

(81)

Hence

10038161003816100381610038161003816119879119860

Ω1205721198912(119910)10038161003816100381610038161003816le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

sdot

infin

sum

119895=1

119895

119908119902 (2119895+1119876)1119902minus120581119901

(82)

Therefore

119868119868 le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

sdot

infin

sum

119895=1

119895119908119902(119876)

1119902minus120581119901

119908119902 (2119895+1119876)1119902minus120581119901


le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

infin

sum

119895=1

119895

(119863119895+1

1)1119902minus120581119901

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

(83)


1

119908119902 (119876)120581119901(int119876

10038161003816100381610038161003816119879119860

Ω120572119891 (119910)

10038161003816100381610038161003816

119902

119908 (119910)119902119889119910)

1119902

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

(84)




119876+ 119891120594

(119876)119888 fl 119891

1+ 119891

2 Then we have

1

119908 (119876)120581119901(int119876

1003816100381610038161003816[119860 119879Ω] 119891 (119910)1003816100381610038161003816

119901119908 (119910) 119889119910)

1119901

le1

119908 (119876)120581119901(int119876

1003816100381610038161003816[119860 119879Ω] 1198911 (119910)1003816100381610038161003816

119901119908 (119910) 119889119910)

1119901

+1

119908 (119876)120581119901(int119876

1003816100381610038161003816[119860 119879Ω] 1198912 (119910)1003816100381610038161003816

119901119908 (119910) 119889119910)

1119901

fl 119868 + 119868119868

(85)


119868 le1

119908 (119876)120581119901

1003817100381710038171003817[119860 119879Ω] 11989111003817100381710038171003817119871119901(119908)

le119862

119908 (119876)120581119901119860lowast

100381710038171003817100381711989111003817100381710038171003817119871119901(119908)

= 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

119908(119876)120581119901

119908 (119876)120581119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(86)


1003816100381610038161003816[119860 119879Ω] 1198912 (119910)1003816100381610038161003816 le

infin

sum

119895=1

int2119895+11198762119895119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899

1003816100381610038161003816119860 (119910) minus 119860 (119911)1003816100381610038161003816

sdot1003816100381610038161003816119891 (119911)

1003816100381610038161003816 119889119911

le 119862

infin

sum

119895=1

1

100381610038161003816100381621198951198761003816100381610038161003816

(int2119895+1119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

119904119889119911)

1119904

sdot (int2119895+1119876

1003816100381610038161003816119860 (119910) minus 119860 (119911)1003816100381610038161003816

1199041015840

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840

le 119862 Ω119871119904

sdot

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

sdot (int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840

+ 119862 Ω119871119904

sdot

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

(int2119895+1119876


10038161003816100381610038161003816

1199041015840

sdot1003816100381610038161003816119891 (119911)

1003816100381610038161003816

1199041015840

119889119911)

11199041015840

fl 1198681198681(119910) + 119868119868

2

(87)




1

119908 (119876)120581119901(int119876

1198681198681(119910)

119901119908 (119910) 119889119910)

1119901

= 119862Ω119871119904

119908 (119876)120581119901

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

(int119876

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

119901

(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

1199011199041015840

119908 (119910) 119889119910)

1119901

le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

119908 (119876)

120581

119901

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

119908 (2119895+1119876)120581119901

119908minus1199041015840

(119901minus1199041015840

)(2119895+1119876)(119901minus1199041015840

)1199011199041015840

sdot (int119876

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

119908 (119876)120581119901

infin

sum

119895=1

1

119908 (2119895+1119876)(1minus120581)119901

(int119876

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

(88)



(int119876

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

le (int119876

1003816100381610038161003816119860 (119910) minus 119898119876119908 (119860)1003816100381610038161003816

119901119908 (119910) 119889119910)

1119901

+10038161003816100381610038161003816119898119876119908 (119860) minus 1198982119895+1119876119908minus119904

1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816119908 (119876)

1119901

fl 119868119868119868 + 119868119881

(89)



Lemma 25 that

119868119868119868 le 119860lowast 119908 (119876)1119901 (90)


10038161003816100381610038161003816119898119876119908 (119860) minus 1198982119895+1119876119908minus119904

1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

le1003816100381610038161003816119898119876119908 (119860) minus 119898119876 (119860)

1003816100381610038161003816 +1003816100381610038161003816119898119876 (119860) minus 1198982119895+1119876 (119860)

1003816100381610038161003816

+100381610038161003816100381610038161198982119895+1119876 (119860) minus 1198982119895+1119876119908minus119904

1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816le

1

119908 (119876)

sdot int119876

1003816100381610038161003816119860 (119905) minus 119898119876 (119860)1003816100381610038161003816 119908 (119905) 119889119905 + 2

119899(119895 + 1) 119860lowast

+1

119908minus1199041015840(119901minus119904

1015840) (2119895+1119876)

sdot int2119895+1119876

1003816100381610038161003816119860 (119905) minus 1198982119895+1119876 (119860)1003816100381610038161003816 119908minus1199041015840

(119901minus1199041015840

)(119905) 119889119905

le 119862 (119895 + 1) 119860lowast

(91)

Hence

119868119881 le 119862 (119895 + 1) 119860lowast 119908 (119876)1119901 (92)

As a result

1

119908 (119876)120581119901(int119876

1198681198681(119910)

119901119908 (119910) 119889119910)

1119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

infin

sum

119895=1

(119895 + 1)119908 (119876)

(1minus120581)119901

119908 (2119895+1119876)(1minus120581)119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(93)


1199011199041015840 we get

1198681198682le 119862 Ω119871119904

sdot

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

119901119908 (119911) 119889119911)

1119901

sdot (int2119895+1119876


10038161003816100381610038161003816

1199011199041015840

(119901minus1199041015840

)

sdot 119908 (119911)minus1199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

le 119862 Ω119871119904 119860lowast

sdot10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

119908 (2119895+1119876)120581119901

sdot 119908minus1199041015840

(119901minus1199041015840

)(2119895+1119876)(119901minus1199041015840

)1199011199041015840

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

sdot

infin

sum

119895=1

1

119908 (2119895+1119876)(1minus120581)119901

(94)

Therefore

1

119908 (119876)120581119901(int119876

119868119868119901

2119908 (119910) 119889119910)

1119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

infin

sum

119895=1

119908 (119876)(1minus120581)119901

119908 (2119895+1119876)(1minus120581)119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(95)




119860119876(119910) = 119860 (119910) minus 119898

119876(nabla119860) 119910 (96)


(119876)119888 fl

1198911+ 119891

2 Then we get

1

119908 (119876)120581119901(int119876

100381610038161003816100381610038161003816119860

Ω119891 (119910)

100381610038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

le1

119908 (119876)120581119901(int119876

100381610038161003816100381610038161003816119860

Ω1198911(119910)100381610038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

+1

119908 (119876)120581119901(int119876

100381610038161003816100381610038161003816119860

Ω1198912(119910)100381610038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

fl 119868 + 119868119868

(97)



119868 le1

119908 (119876)120581119901

100381710038171003817100381710038171003817119860

Ω1198911

100381710038171003817100381710038171003817119871119901(119908)

le119862


100381710038171003817100381711989111003817100381710038171003817119871119901(119908)

le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

119908(119876)120581119901

119908 (119876)120581119901

le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(98)


119901 Ω isin 119871




Competing Interests


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

infin

sum

119895=1

119895

(119863119895+1

1)1119902minus120581119901

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

(83)


1

119908119902 (119876)120581119901(int119876

10038161003816100381610038161003816119879119860

Ω120572119891 (119910)

10038161003816100381610038161003816

119902

119908 (119910)119902119889119910)

1119902

le 119862 sum

|120574|=119898minus1

10038171003817100381710038171198631205741198601003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)

(84)




119876+ 119891120594

(119876)119888 fl 119891

1+ 119891

2 Then we have

1

119908 (119876)120581119901(int119876

1003816100381610038161003816[119860 119879Ω] 119891 (119910)1003816100381610038161003816

119901119908 (119910) 119889119910)

1119901

le1

119908 (119876)120581119901(int119876

1003816100381610038161003816[119860 119879Ω] 1198911 (119910)1003816100381610038161003816

119901119908 (119910) 119889119910)

1119901

+1

119908 (119876)120581119901(int119876

1003816100381610038161003816[119860 119879Ω] 1198912 (119910)1003816100381610038161003816

119901119908 (119910) 119889119910)

1119901

fl 119868 + 119868119868

(85)


119868 le1

119908 (119876)120581119901

1003817100381710038171003817[119860 119879Ω] 11989111003817100381710038171003817119871119901(119908)

le119862

119908 (119876)120581119901119860lowast

100381710038171003817100381711989111003817100381710038171003817119871119901(119908)

= 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

119908(119876)120581119901

119908 (119876)120581119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(86)


1003816100381610038161003816[119860 119879Ω] 1198912 (119910)1003816100381610038161003816 le

infin

sum

119895=1

int2119895+11198762119895119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

1003816100381610038161003816119910 minus 1199111003816100381610038161003816

119899

1003816100381610038161003816119860 (119910) minus 119860 (119911)1003816100381610038161003816

sdot1003816100381610038161003816119891 (119911)

1003816100381610038161003816 119889119911

le 119862

infin

sum

119895=1

1

100381610038161003816100381621198951198761003816100381610038161003816

(int2119895+1119876

1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816

119904119889119911)

1119904

sdot (int2119895+1119876

1003816100381610038161003816119860 (119910) minus 119860 (119911)1003816100381610038161003816

1199041015840

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840

le 119862 Ω119871119904

sdot

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

sdot (int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

11199041015840

+ 119862 Ω119871119904

sdot

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

(int2119895+1119876


10038161003816100381610038161003816

1199041015840

sdot1003816100381610038161003816119891 (119911)

1003816100381610038161003816

1199041015840

119889119911)

11199041015840

fl 1198681198681(119910) + 119868119868

2

(87)




1

119908 (119876)120581119901(int119876

1198681198681(119910)

119901119908 (119910) 119889119910)

1119901

= 119862Ω119871119904

119908 (119876)120581119901

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

(int119876

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

119901

(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

1199041015840

119889119911)

1199011199041015840

119908 (119910) 119889119910)

1119901

le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

119908 (119876)

120581

119901

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

119908 (2119895+1119876)120581119901

119908minus1199041015840

(119901minus1199041015840

)(2119895+1119876)(119901minus1199041015840

)1199011199041015840

sdot (int119876

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

119908 (119876)120581119901

infin

sum

119895=1

1

119908 (2119895+1119876)(1minus120581)119901

(int119876

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

(88)



(int119876

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

le (int119876

1003816100381610038161003816119860 (119910) minus 119898119876119908 (119860)1003816100381610038161003816

119901119908 (119910) 119889119910)

1119901

+10038161003816100381610038161003816119898119876119908 (119860) minus 1198982119895+1119876119908minus119904

1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816119908 (119876)

1119901

fl 119868119868119868 + 119868119881

(89)



Lemma 25 that

119868119868119868 le 119860lowast 119908 (119876)1119901 (90)


10038161003816100381610038161003816119898119876119908 (119860) minus 1198982119895+1119876119908minus119904

1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

le1003816100381610038161003816119898119876119908 (119860) minus 119898119876 (119860)

1003816100381610038161003816 +1003816100381610038161003816119898119876 (119860) minus 1198982119895+1119876 (119860)

1003816100381610038161003816

+100381610038161003816100381610038161198982119895+1119876 (119860) minus 1198982119895+1119876119908minus119904

1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816le

1

119908 (119876)

sdot int119876

1003816100381610038161003816119860 (119905) minus 119898119876 (119860)1003816100381610038161003816 119908 (119905) 119889119905 + 2

119899(119895 + 1) 119860lowast

+1

119908minus1199041015840(119901minus119904

1015840) (2119895+1119876)

sdot int2119895+1119876

1003816100381610038161003816119860 (119905) minus 1198982119895+1119876 (119860)1003816100381610038161003816 119908minus1199041015840

(119901minus1199041015840

)(119905) 119889119905

le 119862 (119895 + 1) 119860lowast

(91)

Hence

119868119881 le 119862 (119895 + 1) 119860lowast 119908 (119876)1119901 (92)

As a result

1

119908 (119876)120581119901(int119876

1198681198681(119910)

119901119908 (119910) 119889119910)

1119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

infin

sum

119895=1

(119895 + 1)119908 (119876)

(1minus120581)119901

119908 (2119895+1119876)(1minus120581)119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(93)


1199011199041015840 we get

1198681198682le 119862 Ω119871119904

sdot

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

119901119908 (119911) 119889119911)

1119901

sdot (int2119895+1119876


10038161003816100381610038161003816

1199011199041015840

(119901minus1199041015840

)

sdot 119908 (119911)minus1199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

le 119862 Ω119871119904 119860lowast

sdot10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

119908 (2119895+1119876)120581119901

sdot 119908minus1199041015840

(119901minus1199041015840

)(2119895+1119876)(119901minus1199041015840

)1199011199041015840

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

sdot

infin

sum

119895=1

1

119908 (2119895+1119876)(1minus120581)119901

(94)

Therefore

1

119908 (119876)120581119901(int119876

119868119868119901

2119908 (119910) 119889119910)

1119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

infin

sum

119895=1

119908 (119876)(1minus120581)119901

119908 (2119895+1119876)(1minus120581)119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(95)




119860119876(119910) = 119860 (119910) minus 119898

119876(nabla119860) 119910 (96)


(119876)119888 fl

1198911+ 119891

2 Then we get

1

119908 (119876)120581119901(int119876

100381610038161003816100381610038161003816119860

Ω119891 (119910)

100381610038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

le1

119908 (119876)120581119901(int119876

100381610038161003816100381610038161003816119860

Ω1198911(119910)100381610038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

+1

119908 (119876)120581119901(int119876

100381610038161003816100381610038161003816119860

Ω1198912(119910)100381610038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

fl 119868 + 119868119868

(97)



119868 le1

119908 (119876)120581119901

100381710038171003817100381710038171003817119860

Ω1198911

100381710038171003817100381710038171003817119871119901(119908)

le119862


100381710038171003817100381711989111003817100381710038171003817119871119901(119908)

le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

119908(119876)120581119901

119908 (119876)120581119901

le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(98)


119901 Ω isin 119871




Competing Interests


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(int119876

10038161003816100381610038161003816119860 (119910) minus 119898

2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

le (int119876

1003816100381610038161003816119860 (119910) minus 119898119876119908 (119860)1003816100381610038161003816

119901119908 (119910) 119889119910)

1119901

+10038161003816100381610038161003816119898119876119908 (119860) minus 1198982119895+1119876119908minus119904

1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816119908 (119876)

1119901

fl 119868119868119868 + 119868119881

(89)



Lemma 25 that

119868119868119868 le 119860lowast 119908 (119876)1119901 (90)


10038161003816100381610038161003816119898119876119908 (119860) minus 1198982119895+1119876119908minus119904

1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816

le1003816100381610038161003816119898119876119908 (119860) minus 119898119876 (119860)

1003816100381610038161003816 +1003816100381610038161003816119898119876 (119860) minus 1198982119895+1119876 (119860)

1003816100381610038161003816

+100381610038161003816100381610038161198982119895+1119876 (119860) minus 1198982119895+1119876119908minus119904

1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816le

1

119908 (119876)

sdot int119876

1003816100381610038161003816119860 (119905) minus 119898119876 (119860)1003816100381610038161003816 119908 (119905) 119889119905 + 2

119899(119895 + 1) 119860lowast

+1

119908minus1199041015840(119901minus119904

1015840) (2119895+1119876)

sdot int2119895+1119876

1003816100381610038161003816119860 (119905) minus 1198982119895+1119876 (119860)1003816100381610038161003816 119908minus1199041015840

(119901minus1199041015840

)(119905) 119889119905

le 119862 (119895 + 1) 119860lowast

(91)

Hence

119868119881 le 119862 (119895 + 1) 119860lowast 119908 (119876)1119901 (92)

As a result

1

119908 (119876)120581119901(int119876

1198681198681(119910)

119901119908 (119910) 119889119910)

1119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

infin

sum

119895=1

(119895 + 1)119908 (119876)

(1minus120581)119901

119908 (2119895+1119876)(1minus120581)119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(93)


1199011199041015840 we get

1198681198682le 119862 Ω119871119904

sdot

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

(int2119895+1119876

1003816100381610038161003816119891 (119911)1003816100381610038161003816

119901119908 (119911) 119889119911)

1119901

sdot (int2119895+1119876


10038161003816100381610038161003816

1199011199041015840

(119901minus1199041015840

)

sdot 119908 (119911)minus1199041015840

(119901minus1199041015840

)119889119911)

(119901minus1199041015840

)1199011199041015840

le 119862 Ω119871119904 119860lowast

sdot10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

infin

sum

119895=1

100381610038161003816100381610038162119895+211987610038161003816100381610038161003816

1119904

100381610038161003816100381621198951198761003816100381610038161003816

119908 (2119895+1119876)120581119901

sdot 119908minus1199041015840

(119901minus1199041015840

)(2119895+1119876)(119901minus1199041015840

)1199011199041015840

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

sdot

infin

sum

119895=1

1

119908 (2119895+1119876)(1minus120581)119901

(94)

Therefore

1

119908 (119876)120581119901(int119876

119868119868119901

2119908 (119910) 119889119910)

1119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

infin

sum

119895=1

119908 (119876)(1minus120581)119901

119908 (2119895+1119876)(1minus120581)119901

le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(95)




119860119876(119910) = 119860 (119910) minus 119898

119876(nabla119860) 119910 (96)


(119876)119888 fl

1198911+ 119891

2 Then we get

1

119908 (119876)120581119901(int119876

100381610038161003816100381610038161003816119860

Ω119891 (119910)

100381610038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

le1

119908 (119876)120581119901(int119876

100381610038161003816100381610038161003816119860

Ω1198911(119910)100381610038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

+1

119908 (119876)120581119901(int119876

100381610038161003816100381610038161003816119860

Ω1198912(119910)100381610038161003816100381610038161003816

119901

119908 (119910) 119889119910)

1119901

fl 119868 + 119868119868

(97)



119868 le1

119908 (119876)120581119901

100381710038171003817100381710038171003817119860

Ω1198911

100381710038171003817100381710038171003817119871119901(119908)

le119862


100381710038171003817100381711989111003817100381710038171003817119871119901(119908)

le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

119908(119876)120581119901

119908 (119876)120581119901

le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(98)


119901 Ω isin 119871




Competing Interests


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119868 le1

119908 (119876)120581119901

100381710038171003817100381710038171003817119860

Ω1198911

100381710038171003817100381710038171003817119871119901(119908)

le119862


100381710038171003817100381711989111003817100381710038171003817119871119901(119908)

le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

119908(119876)120581119901

119908 (119876)120581119901

le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(98)


119901 Ω isin 119871




Competing Interests


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article On the Theory of Multilinear Singular ...downloads.hindawi.com/journals/jfs/2016/4149314.pdf · some classical singular integrals in harmonic analysis on them, such