repository.enu.kzrepository.enu.kz/bitstream/handle/123456789/1130/Convolution... · J Fourier Anal Appl (2011) 17:486–505 DOI 10.1007/s00041-010-9159-9 Convolution Inequalities

J Fourier Anal Appl (2011) 17:486–505DOI 10.1007/s00041-010-9159-9

Convolution Inequalities in Lorentz Spaces

Erlan Nursultanov · Sergey Tikhonov

Received: 21 November 2009 / Revised: 14 August 2010 / Published online: 30 November 2010© Springer Science+Business Media, LLC 2010

Abstract In this paper we study boundedness of the convolution operator in differ-ent Lorentz spaces. We obtain the limit case of the Young-O’Neil inequality in theclassical Lorentz spaces. We also investigate the convolution operator in the weightedLorentz spaces.

Keywords Young-O’Neil inequality · Lorentz spaces · Convolution

Mathematics Subject Classification (2000) Primary 44A35 · Secondary 46E30 ·47G10

1 Introduction

The Young convolution inequality of the form

‖K ∗ f ‖p ≤ ‖f ‖p ‖K‖1

and in a more general form

‖K ∗ f ‖q ≤ ‖f ‖p‖K‖r , 1 + 1

q= 1

p+ 1

r, 1 ≤ p, r, q ≤ ∞ (1.1)

Communicated by Hans G. Feichtinger.

This paper is in final form and no version of it will be submitted for publication elsewhere.

E. NursultanovGumilyov Eurasian National University, Munatpasova 7, 010010 Astana, Kazakhstane-mail: [email protected]

S. Tikhonov (�)ICREA and Centre de Recerca Matemàtica (CRM), 08193, Bellaterra, Barcelona, Spaine-mail: [email protected]

mailto:[email protected]

mailto:[email protected]

J Fourier Anal Appl (2011) 17:486–505 487

plays a very important role both in Harmonic Analysis and PDE (see, e.g., [1], [5,Chap. 4, Sects. 2, 4], [8], [21, Chap. V, Sect. 1]). A sharp constant in (1.1) was foundin [3]. An analogue of Young’s inequality (1.1) in weighted Lebesgue spaces Lp(ω)

and Wiener amalgam spaces W(Lp,Lq(ω)) was investigated in [9, 12] (see refer-ences therein). In [16], the authors studied sufficient and necessary conditions on ω

to obtain the following convolution inequality in Lp(ω)

‖K ∗ f ‖Lp(ω) ≤ c ‖f ‖Lp(ω)‖K‖Lp(ω), 1 ≤ p ≤ ∞.

Moreover, O’Neil [18], Hunt [13], Yap [22], and Blozinski [6] studied the bound-edness of the convolution operator A given by

Af (y) =∫

D

K(y − x)f (x)dx (1.2)

for functions K and f belonging to suitable Lorentz spaces and where the domain D

is Rn.

In particular, the following Young-O’Neil inequality was obtained: for 1 <

p,q, r < ∞, 0 < h1, h2, h3 ≤ ∞, 1 + 1q

= 1p

+ 1r, and 1

h1= 1

h2+ 1

h3, one has

‖K ∗ f ‖Lq,h1 (�) ≤ c ‖f ‖Lp,h2 (D)‖K‖Lr,h3 (�−D), (1.3)

where � − D = {x − y : x ∈ �,y ∈ D}. We note that inequality (1.3) unlike (1.1)gives the Hardy-Littlewood fractional integration theorem, which corresponds to themodel case in which K(x) = |x|−1/r .

The limiting case of inequality (1.3), that is, when 1 = 1p

+ 1r, 1 < p < ∞, 1

h1=

1h2

+ 1h3

< 1, and 1 ≤ h2, h3 ≤ ∞ was investigated in [8]:

‖K ∗ f ‖BWh1≤ c ‖f ‖Lp,h2 (Rn)

(‖K‖Lr,h3 (Rn) + ‖K‖L1(Rn)

), (1.4)

where c = c(p,h2, h3) and

‖ψ‖BWh1=

(∫ 1

0

(ψ∗(t)

1 + | log t |)h1 dt

t

)1/h1

.

Young-O’Neil inequality was also studied for weighted Lorentz spaces. Ker-man [15] proved the analogue of (1.3) for power weights. It is worth mentioningthat in the case of non-homogeneous measures, operator (1.2) does not satisfy allrequirements from [18] and needs thorough investigation.

In this paper we continue investigating the Young-type inequalities in differentLorentz spaces. The outline of the paper is as follows. In Sect. 2 we study the bound-edness of the operator A from Lp,h2(�) into Lp,h1(�), i.e., the limit case of theYoung-O’Neil inequality (p = q and r = 1). It is known (see [7, Theorem 2]) that if� = R

n, h1 < h2 ≤ ∞ and K ≥ 0, then

A : Lp,h2(Rn) −→ Lp,h1(R

n)

488 J Fourier Anal Appl (2011) 17:486–505

implies A ≡ 0 and in turn Ka.e.= 0. In contrast we show that in the case when � has

finite measure, the same problem has a nontrivial solution: for 1 ≤ p = q < ∞ weprove (see Theorem 2.1)

‖(K ∗ f )∗∗‖p,h1 ≤ c ‖f ∗∗‖p,h2‖K∗∗‖1,h3 ,1

h1= 1

h2+ 1

h3, (1.5)

where

ψ∗∗(x) = 1

x

∫ x

0ψ∗(t)dt (1.6)

and ψ∗ is the decreasing rearrangement of ψ . Moreover, we show that the factor‖K∗∗‖L1,h3

in (1.5) could not be changed to either ‖K‖L1,h3, or ‖K∗∗‖L1,h

for anyh > h3. Note that max(‖K‖L1,h3

,‖K∗∗‖L1,h) ≤ c‖K∗∗‖1,h3 .

We mention that the key point in the proof of the classical estimate (1.3) is thefollowing convolution inequality (see [18]):

(K ∗ f )∗∗(t) ≤ tK∗∗(t)f ∗∗(t) +∫ ∞

t

K∗(u)f ∗(u) du.

Our main idea is to use a different version of the above convolution inequality, namelywith K∗∗ − K∗ on the right-hand side in place of K∗ (see Lemma 2.2).

The case p = q = ∞ is studied separately in Sect. 3; here the proper setting isgiven by L∞,q -spaces introduced in [4] and [2] and we obtain (see Theorem 3.1)

‖K ∗ f ‖L∞,h1≤ 2‖f ‖L∞,h2

‖K∗∗‖L1,h3.

Note that since L∞,h1 ⊂ BWh1 with h1 > 1 (see [2, Th. 3.1]), this gives the limitingcase of (1.4):

‖K ∗ f ‖BWh1≤ c ‖f ‖L∞,h2

‖K∗∗‖L1,h3.

We show (see Theorem 3.2) that the constant c can be taken as 2h′1.

Section 4 provides a Young-O’Neil-type inequality in Lorentz spaces with weightfunction ω(t), namely �q(ω) and �q(ω), where

�q(ω) ={

‖f ‖�q(ω) =(∫ ∞

0

(f ∗(t)

)qω(t) dt

)1/q

< ∞}

and �q(ω) is defined similarly with f ∗∗ in place of f ∗ (see, e.g., [19]). Finally wemention that some results of this paper, namely Theorems 2.1 and 3.1, were an-nounced in the note [17].

By C,Ci, c we will denote positive constants that may be different in differentcontexts and depend only on parameters pi, qi, and hi . We will write F � G if F ≤C1G and G ≤ C2F .


2 Convolution in the Lorentz Space of Periodic Functions: the Case of1 ≤ p < ∞

Let Lp,q [0,1] be the Lorentz space of all 1-periodic functions with the quasi-normgiven by

‖f ‖p,q = ‖f ‖Lp,q [0,1] =(∫ 1

0(t

1p f ∗(t))q dt

t

) 1q

.

It is known [5, p. 219] that for 1 < p < ∞ and 1 ≤ q ≤ ∞ we have

‖f ‖Lp,q ≤ ‖f ∗∗‖p,q :=(∫ 1

0(t

1p f ∗∗(t))q dt

t

) 1q ≤ p′‖f ‖Lp,q , (2.1)

where f ∗∗ is defined by (1.6) and can be written as [5, p. 53]

f ∗∗(t) = sup|e|=t

e⊂[0,1]

1

|e|∫

e

|f (x)|dx. (2.2)

Here p′ = pp−1 and |e| := μe.

Note that for p = 1, the norms ‖f ‖Lp,q and ‖f ∗∗‖p,q are not equivalent. Fora fixed f ∈ L1 the functional ‖f ∗∗‖1,q is non-decreasing as a function of 1/q ∈[0,+∞).

We will need the following lemmas.

Lemma 2.1 Let f (t) be a non-increasing non-negative function on (0,∞), 0 ≤φ(t) ≤ 1 and

∫ ∞0 φ(t)dt = d < ∞. Then

∫ d

0f (t)dt ≥

∫ ∞

0f (t)φ(t)dt. (2.3)

Proof Indeed, inequality (2.3) follows from Hardy’s lemma (see, e.g., [5, Prop. 3.6,p. 56]):

∫ x

0 h1 dt ≥ ∫ x

0 h2 dt implies∫ ∞

0 h1f dt ≥ ∫ ∞0 h2f dt for any non-increasing

nonnegative function f on (0,∞). �

Lemma 2.2 Let f,g, and K be measurable 1-periodic functions and g is non-negative. Then

∫ 1

0g(t)(K ∗ f )∗∗(t)dt ≤ ‖g‖L1‖f ‖L1‖K‖L1

+∫ 1

0f ∗∗(t)

(K∗∗(t) − K∗(t)

)(∫ t

0g(s)ds

)dt. (2.4)

Proof Let us assume the right-hand side of (2.4) is finite. From (2.2) and the Hardy-Littlewood inequality [5, p. 44], we write


∫ 1

0g(s)(K ∗ f )∗∗(s)ds

≤∫ 1

0g(s) sup

|e|=se⊂[0,1]

∫ 1

0|f (x)| 1

|e|∫

e

|K(y − x)|dy dx ds

≤∫ 1

0g(s) sup

|e|=se⊂[0,1]

∫ 1

0f ∗(t)

(1

|e|∫

e

|K(y − ·)|dy

)∗∗(t) dt ds

=∫ 1

0g(s) sup

|e|=se⊂[0,1]

∫ 1

0f ∗(t) sup

|ω|=tω⊂[0,1]

1

|e|1

|ω|∫

e

∫ω

|K(y − x)|dx dy dt ds

≤∫ 1

0g(s)

∫ 1

0f ∗(t)

⎛⎝ sup

|e|=se⊂[0,1]

sup|ω|=tω⊂[0,1]

1

|e|1

|ω|∫

e

∫ω

|K(y − x)|dxdy

⎞⎠dt ds.

We consider

�(s, t) = sup|e|=s|ω|=t

1

|e|1

|ω|∫

ω

∫e

|K(y − x)|dy dx

= sup|e|=s|ω|=t

1

|e|1

|w|∫ 1

0χe(x)

∫ 1

0χω(y) |K(y − x)|dy dx

= sup|e|=s|w|=t

1

|e|1

|w|∫ 1

0|K(x)| |e ∩ (w + x)|dx

≤ sup|e|=s|w|=t

1

|e|1

|w|∫ 1

0K∗(ξ)φ(ξ)dξ,

where φ(ξ) is the decreasing rearrangement of the function |e ∩ (ω + x)|.Then φ(ξ) satisfies φ(s) ≤ min(|e|, |w|) and

∫ 1

0φ(ξ)dξ = |e||w|.

We assume that s = |e| < |w| = t , then the function φ0(ξ) = φ(ξ)/|e| satisfies con-ditions of Lemma 2.1 with d = |w|. Then for s < t we have

�(s, t) ≤ sup|e|=s,e⊂[0,1]|w|=t,w⊂[0,1]

1

|w|∫ 1

0K∗(ξ)φ0(ξ)dξ

≤ sup|w|=t

1

|w|∫ |w|

0K∗(ξ)dξ = K∗∗(t).


As above, for s ≥ t we write

�(s, t) ≤ sup|e|=s

1

|e|∫ |e|

0K∗(ξ)dξ = K∗∗(s).

Combining these estimates, we have

∫ 1

0g(s)(K ∗ f )∗∗(s)ds ≤

∫ 1

0g(s)

∫ 1

0f ∗(t)K∗∗(max{s, t})dt ds. (2.5)

Using

g(s)

∫ s

0f ∗(t)dt + f ∗(s)

∫ s

0g(s)dt =

(∫ s

0g(t)dt

∫ s

0f ∗(t)dt

)′,

and inequality (2.5), we get

∫ 1

0g(s)(K ∗ f )∗∗(s)ds ≤

∫ 1

0g(s)

∫ 1

0f ∗(t)K∗∗(max(t, s))dt ds

=∫ 1

0K∗∗(s)

(g(s)

∫ s

0f ∗(t)dt + f ∗(s)

∫ s

0g(t)dt

)ds

=∫ 1

0K∗∗(s)

(∫ s

0g(t)dt

∫ s

0f ∗(t)dt

)′ds.

Integrating by parts, we get for ε > 0

∫ 1

ε

g(s)(K ∗ f )∗∗(s)ds

≤ K∗∗(s)∫ s

0f ∗(t)dt

∫ s

0g(t)dt

∣∣∣∣1

ε

−∫ 1

ε

(∫ s

0f ∗(t)dt

∫ s

0g(t)dt

)(K∗∗(s))′ds

≤ K∗∗(1)

∫ 1

0f ∗(t)dt

∫ 1

0g(t)dt +

∣∣∣∣∫ 1

0

(∫ s

0f ∗(t)dt

∫ s

0g(t)dt

)(K∗∗(s))′ds

∣∣∣∣.Hence, using

(K∗∗(s))′ = −1

s

(K∗∗(s) − K∗(s)

), (2.6)

we obtain∫ 1

0g(t)(K ∗ f )∗∗(t)dt ≤ ‖g‖L1‖f ‖L1‖K‖L1 +

∫ 1

0

(∫ t

0g(s)ds

)f ∗∗(t)

(K∗∗(t)

− K∗(t))dt.

The proof is now complete. �


Now we give an elementary proof of the crucial convolution lemma of O’Neil-type(cf. [18, Lemma 1.5]) for the case of periodic functions.

Lemma 2.3 Suppose f and K are 1-periodic integrable functions, then

(K ∗ f )∗∗(t) ≤ f ∗∗(t)∫ t

0K∗(s)ds +

∫ 1

t

f ∗∗(s)K∗(s)ds. (2.7)

Proof Indeed, by (2.2), we estimate

(K ∗ f )∗∗(t) = (f ∗ K)∗∗(t) ≤ sup|e|=t

∫ 1

0|K(x)| 1

|e|∫

e

|f (y − x)| dy dx.

Further, using the same calculations as in the proof of Lemma 2.2, we get

sup|e|=t

∫ 1

0|K(x)| 1

|e|∫

e

|f (y − x)| dy dx

≤ sup|e|=t

∫ 1

0K∗(s) sup

|w|=s

1

|e||w|∫

w

∫e

|f (y − x)|dy dx ds

≤∫ 1

0K∗(s)f ∗∗(max(t, s))ds

= f ∗∗(t)∫ t

0K∗(s)ds +

∫ 1

t

K∗(s))f ∗∗(s)ds,

which concludes the proof. �

Lemma 2.4 Let 1 < p < ∞ and 1 ≤ q < ∞. Then

‖f ‖Lp,q = supg∈Dp′,q′

∫ 1

0g(t)f ∗(t)dt,

where

Dp′,q ′ ={

g : g ≥ 0,

(∫ 1

0

(t

1p′ g(t)

)q ′dt

t

)1/q ′

= 1

}.

Proof Indeed,

∫ 1

0g(t)f ∗(t)dt ≤

(∫ 1

0

(t

1p f ∗(t)

)q dt

t

)1/q (∫ 1

0

(t

1p′ g(t)

)q ′dt

t

)1/q ′

.

On the other hand, we define

g0(t) = tqp

−1(f ∗(t))q−1

(∫ 10 (t

1p f ∗(t))q dt

t

)1/q ′ .


Hence, g0 ∈ Dp′q ′ and∫ 1

0g0(t)f

∗(t)dt = ‖f ‖Lp,q .

�

We are now in a position to present the limiting case of Young-O’Neil’s convolu-tion inequality.

Theorem 2.1 Let 1 ≤ p < ∞, 1 ≤ h1, h2, h3 ≤ ∞, and

1

h1= 1

h2+ 1

h3. (2.8)

Suppose K is a 1-periodic function such that K∗∗ ∈ L1,h3 [0,1].a) If 1 < p < ∞ and f ∈ Lp,h2 [0,1], then K ∗ f ∈ Lp,h1 and

‖K ∗ f ‖Lp,h1≤ c ‖f ‖Lp,h2

‖K∗∗‖L1,h3. (2.9)

b) If p = 1 and f ∗∗ ∈ L1,h2 [0,1], then (K ∗ f )∗∗ ∈ L1,h1 and

‖(K ∗ f )∗∗‖L1,h1≤ c ‖f ∗∗‖L1,h2

‖K∗∗‖L1,h3.

Proof a). Let 1 < p < ∞, 1 ≤ h1 < ∞. By Lemmas 2.4 and 2.2, we have

‖f ∗ K‖Lp,h1

= supg∈Dp′,h′

1

∫ 1

0g(t)(K ∗ f )∗(t)dt ≤ sup

g∈Dp′,h′1

∫ 1

0g(t)(K ∗ f )∗∗(t)dt

≤ supg∈Dp′,h′

1

(‖g‖L1‖f ‖L1‖K‖L1 +

∫ 1

0

(K∗∗(t) − K∗(t)

)f ∗∗(t)

∫ t

0g(s)dsdt

)

Applying Hölder’s inequality (cf. (2.8)),

‖f ∗ K‖Lp,h1≤ sup

g∈Dp′,h′1

(‖g‖L1‖f ‖L1‖K‖L1

+∫ 1

0

(t1− 1

h3 K∗∗(t))(

t1p

− 1h2 f ∗∗(t)

)(t

1p′ − 1

h′1−1

∫ t

0g(s)ds

)dt

)

≤ c

(‖f ‖L1‖K‖L1 + sup

g∈Dp′,h′1

‖K∗∗‖1,h3‖f ∗∗‖p,h2

×(∫ 1

0

(t

1p′ −1

∫ t

0g(s)ds

)h′1 dt

t

)1/h′1)

. (2.10)


Since 1 < p < ∞, then (2.1) yields ‖f ∗∗‖Lp,h2� ‖f ‖Lp,h2

and Hardy’s inequalitygives

(∫ 1

0

(t

1p′ −1

∫ t

0g(s)ds

)h′1 dt

t

)1/h′1

≤ c

(∫ 1

0

(t

1p′ g(t)

)h′1 dt

t

)1/h′1

.

Therefore, (2.10) implies

‖(f ∗ K)‖Lp,h1≤ c ‖K∗∗‖L1,h3

‖f ‖Lp,h2.

Let now h1 = ∞. Then using Lemma 2.3 we have

‖f ∗ K‖Lp,∞ ≤ supt

t1/p(f ∗ K)∗∗(t)

≤ supt

t1/pf ∗∗(t)∫ t

0K∗(s)ds + sup

tt1/p

∫ 1

t

f ∗∗(s)K∗(s)ds

≤ 2‖K‖1 supt

t1/pf ∗∗(t) ≤ 2p′ ‖K‖1 supt

t1/pf ∗(t)

= 2p′ ‖K∗∗‖1,∞‖f ‖p,∞.

b). Let p = 1 and 1 < h1 < ∞. Then using [10, Th 4.1] and periodicity of func-tions, we get1

‖(K ∗ f )∗∗‖1,h1 � sup‖g‖L∞,h′

1=1

∫ 1

0g(y)(K ∗ f )(y)dy

= sup‖g‖L∞,h′

1=1

∫ 1

0f (x)(g ∗ K)(x)dx

≤ sup‖g‖L∞,h′

1=1

∫ 1

0f ∗(s)(g ∗ K)∗∗(s)ds.

We again apply Lemma 2.2 and the Hölder inequality:

‖(K ∗ f )∗∗‖1,h1

≤ c sup‖g‖L∞,h′

1=1

(‖f ‖L1‖g‖L1‖K‖L1 +

∫ 1

0tf ∗∗(t)K∗∗(t)

(g∗∗(t) − g∗(t)

)dt

)

≤ c sup‖g‖L∞,h′

1=1

‖f ∗∗‖1,h2‖K∗∗‖1,h3

⎛⎝‖g‖1 +

(∫ 1

0

(g∗∗ − g∗)h′1

tdt

) 1h′

1

⎞⎠

= c ‖f ∗∗‖1,h2‖K∗∗‖1,h3 .

1See the definition of the space L∞,h′1

in the next section.


Let now p = h1 = 1. Then we use Lemma 2.3:

‖(K ∗ f )∗∗‖1,1 ≤∫ 1

0tf ∗∗(t)K∗∗(t)dt +

∫ 1

0

∫ 1

t

f ∗∗(s)K∗(s)ds dt

≤ 2∫ 1

0tf ∗∗(t)K∗∗(t)dt ≤ 2‖f ∗∗‖1,h2‖K∗∗‖1,h3 ,

where h3 = h′2. We remark that in this case if the right-hand side is finite, then K ∗ f

belongs to the space LlogL.Finally, if p = 1 and h1 = ∞, then by Fubini’s theorem,

‖(K ∗ f )∗∗‖1,∞ = ‖K ∗ f ‖1 ≤ ‖K‖1‖f ‖1 = ‖K∗∗‖1,∞‖f ∗∗‖1,∞.

The proof is complete. �

Let us give two examples showing the sharpness of the results of Theorem 2.1.First, we prove that in inequality (2.9), the factor ‖K∗∗‖L1,h3

could not be changed to

‖K‖L1,h3. That is, in general, for 1 ≤ p = q < ∞ and 1

h1= 1

h2+ 1

h3the inequality

‖K ∗ f ‖Lp,h1≤ C‖f ‖Lp,h2

‖K‖L1,h3(2.11)

does not hold.

Example 2.1 Let 1 ≤ p < ∞,N ∈ N,N > 4(ep + 1). We define f (t) =(min(N,1/t))1/p and K(t) = min(N,1/t). Then

‖f ‖Lp,h2� (1 + lnN)1/h2

and

‖K‖L1,h3� (1 + lnN)1/h3 = (1 + lnN)1/h1−1/h2 .

To prove that (2.11) is not true, let us first define

ϕ(x) ={

0 for x ∈ [0, a)

(K ∗ f )(x) for x ∈ [a,1), a = 1

N

(ep + 1

).

Then if x ∈ [a,1) we have

ϕ(x) ≥∫ x− 1

N

1N

ds

s1p (x − s)

≥ 1

(x − 1N

)1p

∫ x− 1N

1N

ds

x − s= N1/p ln(Nx − 1)

(Nx − 1)1/p.

Noting (ln ξ

ξ1/p )′ = p−ln ξ

pξ1/p+1 < 0 for ξ ∈ (ep,1], we obtain that the function ln ξ

ξ1/p is de-

creasing on [ep,1]. Hence,

ϕ∗(t) ≥ ln(N(t + a) − 1)(t + a − 1

N

)1/pfor t ∈ (0,1 − a).


Using this, we get

‖f ∗ K‖h1Lp,h1

≥∫ 1

0(t1/pϕ∗(t))h1

dt

t≥

∫ 1−a

0

(t1/p ln(N(t + a) − 1)

(t + a − 1N

)1/p

)h1dt

t

≥∫ 1−a

a

(t1/p ln(N(t + a) − 1)

(t + a − 1N

)1/p

)h1dt

t

≥ 2− h1p

∫ 1−a

a

(ln(N(t + a) − 1)

)h1 dt

t

� lnh1+1 (N(t + a) − 1

) |1−aa � (1 + lnN)h1+1.

Thus, (2.11) implies

lnN ≤ C.

This contradiction concludes the proof.We now provide an example showing sharpness of (2.9) in the following sense: if

1 ≤ p = q < ∞ and 1h1

= 1h2

+ 1h3

, then for any h > h3 the inequality

‖K ∗ f ‖Lp,h1≤ C‖f ‖Lp,h2

‖K∗∗‖L1,h

does not hold.

Example 2.2 Let 1 < p < ∞,1 ≤ h1, h2 ≤ ∞ and 1/h3 = (1/h1 − 1/h2)+. Thenfor the Cesàro operator of 1-periodic function f , given by

σN(f ;x) = 1

N + 1

N∑k=0

Sk(f ;x) =∫ 1

0f (y)FN(x − y)dy, N ∈ N,

FN(x) = 1

N + 1

sin2 π(N + 1)x

sin2 πx,

we claim

‖σN‖Lp,h2→Lp,h1� ‖F ∗∗

N ‖L1,h3� (1 + lnN)1/h3 .

As usual, Sk(f, x) denotes the k-th partial sum of the Fourier series of f .

Proof of the claim By Theorem 2.1, we have

‖σN‖Lp,h2→Lp,h1≤ C‖F ∗∗

N ‖L1,h3.


Further, we estimate

‖F ∗∗N ‖L1,h3

≤(∫ 1/(N+1)

0

(∫ t

0F ∗

N(s)ds

)h3 dt

t

+∫ 1

1/(N+1)

((∫ 1/(N+1)

0+

∫ t

1/(N+1)

)F ∗

N(s)ds

)h3dt

t

)1/h3

.

Using the known inequality (see, e.g., [23, Chap. 3, (3.10)])

F ∗N(s) ≤ c min

((N + 1),

1

(N + 1)s2

), s ∈ (0,1],

we write

‖F ∗∗N ‖L1,h3

≤ c

((N + 1)h3

∫ 1/(N+1)

0th3−1dt + 2

∫ 1

1/(N+1)

dt

t

)1/h3

≤ c (1 + lnN)1/h3 .

Thus, we get

‖σN‖Lp,h2→Lp,h1≤ c (1 + lnN)1/h3 .

On the other hand, defining

f0(x) =N∑

k=1

1

k1/pe2πikx

and using Hardy-Littlewood’s theorem on monotone Fourier coefficients for theLorentz space [20], we obtain

‖f0‖Lp,h2�

(N∑

k=1

(k1/p 1

k1/p

)h2 1

k

)1/h2

� (1 + lnN)1/h2

‖σN(f0)‖Lp,h1�

(N∑

k=1

(N − k

N

)h1 1

k

)1/h1

� (1 + lnN)1/h1 .

Therefore, we derive

‖σN‖Lp,h2→Lp,h1≥ ‖σN(f0)‖Lp,h1

‖f0‖Lp,h2

� (1 + lnN)1/h1−1/h2 = (1 + lnN)1/h3 ,

completing the proof. �


3 Convolution in the Lorentz Space of Periodic Functions: the Case of p = ∞Theorem 2.1 does not encompass the limit case p = ∞ (hi < ∞). It is clear that inthis case the classical Lorentz space is trivial. We consider another scale of Lorentzspaces.

Following Bennett et al. [4] (see also [2], [10]), we define L∞,q [0,1] as follows

L∞,q [0,1] ={

f ∈ L1[0,1] : ‖f ‖L∞,q [0,1]

:= ‖f ‖L1[0,1] +(∫ 1

0

(f ∗∗ − f ∗)q dt

t

) 1q

< ∞}

.

We remark that for any q, s ∈ [1,∞] and p ∈ [1,∞), one has L∞,q [0,1] ↪→Lp,s[0,1]. What is more, the following embedding holds: for 1 ≤ p < ∞ and1 ≤ q < q1 ≤ ∞

L∞[0,1] = L∞,1[0,1] ↪→ L∞,q [0,1] ↪→ L∞,q1 [0,1] ↪→ Lp[0,1]. (3.1)

Indeed, for 1 ≤ q < q1 < ∞ we get(∫ 1

0

(f ∗∗ − f ∗)q1 dt

t

)1/q1

=( ∞∑

ν=0

∫ 2−ν

2−ν−1

(f ∗∗ − f ∗)q1 dt

t

)1/q1

≤( ∞∑

ν=0

(∫ 2−ν

2−ν−1

(f ∗∗ − f ∗)q1 dt

t

)q/q1)1/q

and using monotonicity of t(f ∗∗(t) − f ∗(t)

),

≤ C

( ∞∑ν=0

(f ∗∗(1/2ν) − f ∗(1/2ν)

)q)1/q

≤ C

(f ∗∗(1)q +

∞∑ν=1

∫ 2−ν+1

2−ν

(f ∗∗ − f ∗)q dt

t

)1/q

≤ C

(‖f ‖1 +

(∫ 1

0

(f ∗∗ − f ∗)q dt

t

)1/q).

For q1 = ∞ the proof is similar. Also,

‖f ‖L∞,1 = ‖f ‖1 +∫ 1

0

(f ∗∗ − f ∗) dt

t= ‖f ‖1 −

∫ 1

0

(f ∗∗(t)

)′dt

= ‖f ‖1 + f ∗∗(0) − f ∗∗(1) = ‖f ‖L∞ ,

i.e., L∞,1 = L∞.The last embedding in (3.1) follows from Hölder’s inequality (one can assume that

p < q1)

‖f ‖Lp = ‖f ‖Lp,p � ‖f ‖L1 +(∫ 1

0

(f ∗∗ − f ∗)p

dt

)1/p


≤ ‖f ‖L1 +(∫ 1

0t

pq1−p dt

)1/p−1/q1(∫ 1

0

(f ∗∗ − f ∗)q1 dt

t

)1/q1

≤ c ‖f ‖L∞,q1.

Theorem 3.1 Let 1 ≤ h1, h2, h3 ≤ ∞ and

0 <1

h1= 1

h2+ 1

h3.

For any 1-periodic functions f and K such that f ∈ L∞,h2 [0,1] and K ∈ L1,h3 [0,1]we have

‖K ∗ f ‖L∞,h1≤ 4‖f ‖L∞,h2

‖K∗∗‖L1,h3. (3.2)

Proof Let us suppose first that h1 > 1. Let f ∈ L∞,h2 [0,1]. Without loss of general-ity, we may assume that K is of class C∞ and then h ≡ K ∗ f is of C∞.

If

‖h‖1 ≥(∫ 1

0

(h∗∗(t) − h∗(t)

)h1 dt

t

)1/h1

then since

‖K‖L1 ≤ (h3)1/h3 ‖K∗∗‖L1,h3

≤ 2‖K∗∗‖L1,h3, (3.3)

we have

‖h‖L∞,h1≤ 2‖h‖1 ≤ 4‖f ‖L∞,h2

‖K∗∗‖L1,h3

and (3.2) is proved. Now suppose the converse holds true, then

‖h‖L∞,h1≤ 2

(∫ 1

0

(h∗∗(t) − h∗(t)

)h1 dt

t

)1/h1

= 2∫ 1

0g(t)

(h∗∗(t) − h∗(t))t

dt

where

g(t) = (h∗∗(t) − h∗(t))h1−1

(∫ 10 (h∗∗(s) − h∗(s))h1 ds

s

)1/h′1.

The function g satisfies the following conditions:

1) g(t) ≥ 0 and limt→+0 g(t) = 0,2) g(1) ≤ 1,

since ‖h‖1 ≤ (∫ 10 (h∗∗(t) − h∗(t))h1 dt

t

)1/h1 ;

3) (∫ 1

0 (g(t))h′1 dt

t)1/h′

1 = 1,4) g ∈ C∞.

By A we denote a collection of all functions satisfying 1)–4). Then

‖h‖L∞,h1≤ 2 sup

g∈A

∫ 1

0g(t)

(h∗∗(t) − h∗(t)

t

)dt.


For g ∈ A, noting that h is bounded and therefore limt→+0 g(t)h∗∗(t) = 0, we get

∫ 1

0g(t)

(h∗∗(t) − h∗(t)

t

)dt = −

∫ 1

0g(t)

(h∗∗(t)

)′dt

= −g(t)h∗∗(t)∣∣∣1

0+

∫ 1

0g′(t)h∗∗(t)dt

= −g(1)h∗∗(1) +∫ 1

0g′(t)h∗∗(t)dt

≤∫ 1

0g′(t)h∗∗(t)dt.

Hence,

‖K ∗ f ‖L∞,h1≤ sup

g∈A

∫ 1

0g′(t)h∗∗(t) dt = sup

g∈A, g′(t)≥0

∫ 1

0g′(t)h∗∗(t) dt. (3.4)

We now use Lemma 2.3:∫ 1

0g′(t)(K ∗ f )∗∗(t)dt

≤∫ 1

0g′(t)

(f ∗∗(t)

∫ t

0K∗(s)ds +

∫ 1

t


)dt

=∫ 1

0

(g′(t)f ∗∗(t)

∫ t

0K∗(s)ds + f ∗∗(t)K∗(t)

∫ t

0g′(s)ds

)dt

=∫ 1

0f ∗∗(t)

(∫ t

0K∗(s)ds

∫ t

0g′(s)ds

)′dt

= f ∗∗(t)∫ t

0K∗(s)ds

∫ t

0g′(s)ds

∣∣∣∣1

0−

∫ 1

0

(f ∗∗(t)

)′∫ t

0K∗(s)ds

∫ t

0g′(s)ds dt

= g(1)‖f ‖L1‖K‖L1 +∫ 1

0

(f ∗∗(t) − f ∗(t)

)K∗∗(t)g(t)dt.

Taking into account that g ∈ A, we use Hölder’s inequality:

‖K ∗ f ‖L∞,h1≤ 2 sup

g∈A

{‖K‖L1‖f ‖L1 +

(∫ 1

0

(f ∗∗(t) − f ∗(t)

)h2 dt

t

)1/h2

×(∫ 1

0

(tK∗∗(t)

)h3 dt

t

)1/h3(∫ 1

0(g(t))h

′1dt

t

)1/h′1}

≤ 2 ‖f ‖L∞,h2‖K∗∗‖L1,h3

,

and (3.2) follows for h1 > 1 and 1 ≤ h2, h3 ≤ ∞.


Finally, let h1 = 1. Then using Lemma 2.3 and (2.6), we get

‖f ∗ K‖L∞,1 = ‖f ∗ K‖L∞ = supt>0

(K ∗ f )∗∗ (t)

≤ supt>0

(f ∗∗(t)

∫ t

0K∗(s)ds +

∫ 1

t


)

= supt>0

(f ∗∗(t)

∫ t

0K∗(s)ds + f ∗∗(s)

(∫ s

0K∗(ξ)dξ

)∣∣∣∣1

t

−∫ 1

t

(f ∗∗(s)

)′(∫ s

0K∗(ξ)dξ

)ds

)

= ‖f ‖L1‖K‖1 +∫ 1

0

(f ∗∗(s) − f ∗(s)

)K∗∗(s)ds.

Thus, Hölder’s inequality completes the proof:

‖f ∗ K‖L∞,1 ≤ ‖f ‖L∞,h2‖K∗∗‖L1,h′

2= ‖f ‖L∞,h2

‖K∗∗‖L1,h3. �

Let us now prove the limiting case of the Brézis-Wainger inequality (1.4). Wewill need the following weighted inequalities of Hardy type. Denote ‖f ‖

Lr(dtt

):=

(∫ 1

0 |f (t)|r dtt)1/r .

Lemma 3.1 (see, e.g., [8, Lemma 1]) Let 1 < r < ∞ and tϕ(t) ∈ Lr(dtt). Then

∥∥∥∥(1 + | ln t |)−1∫ 1

t

ϕ(s)ds

∥∥∥∥Lr(

dtt

)

≤ r ′ ∥∥tϕ(t)∥∥

Lr(dtt

).

Lemma 3.2 Let 1 < r < ∞ and g ∈ Lr(dtt). Then

∥∥∥∥∫ t

0g(s)

ds

s(1 + | ln s|)∥∥∥∥

Lr(dtt

)

≤ r∥∥g

∥∥Lr(

dtt

).

Proof Using duality arguments, we have

∥∥∥∥∫ t

0g(s)

ds

s(1 + | ln s|)∥∥∥∥

Lr(dtt

)

= sup‖ϕ‖

Lr′ ( dt

t )=1

∫ 1

0

(∫ s

0g(t)

dt

t (1 + | ln t |))

ϕ(s)ds

s

= sup‖ϕ‖

Lr′ ( dt

t )=1

∫ 1

0g(t)

((1 + | ln t |)−1

∫ 1

t

ϕ(s)

sds

)dt

t,


and, by Hölder’s inequality,

≤ sup‖ϕ‖

Lr′ ( dt

t )=1

(∫ 1

0(g(t))r

dt

t

)1/r(∫ 1

0

((1 + | ln t |)−1

∫ 1

t

ϕ(s)

sds

)r ′dt

t

)1/r ′

.

Lemma 3.1 completes the proof. �

Theorem 3.2 Let 1 < h1 < ∞, 1 ≤ h2, h3 ≤ ∞, and 1h1

= 1h2

+ 1h3

. Suppose f ∈L∞,h2 and K∗∗ ∈ L1,h3; then (f ∗K)∗(t)

1+| ln t | ∈ Lh1(dtt) and

∥∥∥∥ (K ∗ f )∗

1 + | ln t |∥∥∥∥

Lh1 ( dtt

)

≤ 2h′1 ‖f ‖L∞,h2

‖K∗∗‖L1,h3. (3.5)

Proof Since ϕ∗(t) ≤ ϕ∗∗(t), we have

∥∥∥∥ (K ∗ f )∗

1 + | ln t |∥∥∥∥

Lh1 ( dtt

)

≤ sup‖g‖

Lh′

1( dt

t )=1

∫ 1

0

g(t)

t (1 + | ln t |) (K ∗ f )∗∗(t)dt.

Then Lemma 2.2 yields

∥∥∥∥ (K ∗ f )∗

1 + | ln t |∥∥∥∥

Lh1 ( dtt

)

≤ sup‖g‖

Lh′

1( dt

t )=1

∥∥∥∥ g(t)

t (1 + | ln t |)∥∥∥∥

L1

‖f ‖L1‖K‖L1

+∫ 1

0

(∫ t

0

g(s)

s(1 + | ln s|)ds

)(f ∗∗(t) − f ∗(t))K∗∗(t)dt.

(3.6)

By Hölder’s inequality and Lemma 3.2,

∥∥∥∥ (K ∗ f )∗

1 + | ln t |∥∥∥∥

Lh1( dt

t )

≤ sup‖g‖

Lh′

1( dt

t )=1

(∥∥∥∥ g(t)

t (1 + | ln t |)∥∥∥∥

L1

‖f ‖L1‖K‖L1

+∥∥∥∥∫ t

0

g(s)

s(1 + | ln s|)ds

∥∥∥∥L

h′1( dt

t )

‖f ∗∗ − f ∗‖Lh2 ( dt

t)‖K∗∗‖L1,h3

)

≤ sup‖g‖

Lh′

1( dt

t )=1

(∥∥∥∥ g(t)

t (1 + | ln t |)∥∥∥∥

L1

‖f ‖L1‖K‖L1

+ h′1 ‖g‖

Lh′1( dt

t)‖f ∗∗ − f ∗‖

Lh2 ( dtt

)‖K∗∗‖L1,h3

).


Finally, taking into account

∥∥∥∥ g(t)

t (1 + | ln t |)∥∥∥∥

L1

≤ (h1 − 1)−1/h1 ‖g‖Lh′

1( dt

t)≤ h′

1‖g‖Lh′

1( dt

t)

and (3.3), we arrive at (3.5). �

Note that using estimate (3.6) it is easy to prove Brézis-Wainger’s inequality (1.4).Indeed,∥∥∥∥ (K ∗ f )∗

1 + | ln t |∥∥∥∥

Lh1 ( dtt

)

≤ sup‖g‖

Lh′

1( dt

t )=1

(∥∥∥∥ g(t)

t (1 + | ln t |)∥∥∥∥

L1

‖f ‖L1‖K‖L1

+∥∥∥∥∫ t

0

g(s)

s(1 + | ln s|)ds

∥∥∥∥L

h′1( dt

t )

(∫ 1

0

(t1/p

(f ∗∗(t) − f ∗(t)

))h2 dt

t

)1/h2

×(∫ 1

0

(t1/p′

K∗∗(t))h3 dt

t

)1/h3)

≤ sup‖g‖

Lh′

1( dt

t )=1

(∥∥∥∥ g(t)

t (1 + | ln t |)∥∥∥∥

L1

‖f ‖L1‖K‖L1 + c‖g‖Lh′

1( dt

t)‖f ‖Lp,h2

‖K‖Lp′,h3

)

≤ c ‖f ‖Lp,h2‖K‖Lp′,h3

.

4 Remarks on Convolution in Weighted Lorentz Spaces

Let us present a convolution estimate of Young-O’Neil type in the weighted Lorentzspaces �q(ω) and �q(ω). Since the proof is similar to the proof of Theorem 2.1, weskip the details.

First, we remark (see [19]) that �q(ω) = �q(ω) if and only if the weight ω satis-fies the Bq -condition, that is,

∫ ∞

x

ω(t)

tqdt ≤ C

xq

∫ x

0ω(t)dt

and we write ω ∈ Bq .

Theorem 4.1 Let ω ∈ Bq and∫ ∞

0 ωdx = ∞. We have

‖K ∗ f ‖�q(ω) � ‖K ∗ f ‖�q(ω) ≤ C‖f ‖�q1 (ω1) ‖K‖�q2 (ω2) (4.1)


for 1/q = 1/q1 + 1/q2, 1 < q < ∞, 1 ≤ q1, q2 ≤ ∞, and

∫ t

0ω(x)dx ≤ c ω(t)

ω1(t)1/q1ω2(t)

1/q2

ω(t)1/q1+1/q2t > 0. (4.2)

Proof The proof follows the line of that of Lemma 2.2 from which we get the fol-lowing inequalities

∫ ∞

0g∗(s)(Af )∗∗(s)ds ≤

∫ ∞

0g∗(s)

∫ ∞

0f ∗(t)K∗∗(max(s, t))dt ds

≤ 2∫ ∞

0tf ∗∗(t)g∗∗(t)K∗∗(t)dt.

Since (4.2) can be rewritten as t ≤ c ω(t)1/q ′ω1(t)

1/q1ω2(t)1/q2 , where

ω(t) = tq′W(t)−q ′

ω(t), W(x) =∫ x

0ω(t)dt,

Hölder’s inequality implies

∫R

g(y)

(∫R

f (x)K(x − y)dx

)dy

≤ 2c

∫ ∞

0

[g∗∗(t)ω(t)1/q ′][

f ∗∗(t)ω1(t)1/q1

][K∗∗(t)ω2(t)

1/q2]dt

≤ 2c ‖g‖�q′

(ω)‖f ‖�q1 (ω1) ‖K‖�q2 (ω2).

It is known from [11, 2.4], [14], and [19, p. 147] that under the conditions ω ∈ Bq

and∫ ∞

0 ω(x)dx = ∞, the dual of �q(ω), 1 < q < ∞, can be identified with �q ′(ω).

Then taking the supremum over all g such that ‖g‖�q′

(ω)= 1, we obtain

‖K ∗ f ‖(�q′

(ω))′ ≤ 2c ‖f ‖�q1 (ω1) ‖K‖�q2 (ω2).

Finally, since ω ∈ Bp , we have(�q(ω)

)′′ = �q(ω) and (4.1) follows. �

Example 4.1 Let ω(t) = tq/h−1ξq

1 (t), ω1(t) = tq1/p−1ξq12 (t), ω2(t) = tq2/r−1ξ

q23 (t),

where 1/q = 1/q1 + 1/q2, 1 + 1/h = 1/p + 1/r , and ξi are slowly oscillating func-tions. Then inequality (4.2) is equivalent to

ξ1(x) ≤ Cξ2(x)ξ3(x),

i.e., in this case we obtain the Young-O’Neil-type inequality for the Lorentz-Zygmund spaces [5, p. 253].

Acknowledgements The authors thank the referees for careful reading of the paper and numerous usefulremarks. The research was partially supported by the Generalitat de Catalunya (2009 SGR 1303), MTM2008-05561-C02-02/MTM, RFFI 09-01-00175, and NSH-3252.2010.1. The paper was started when theauthors were staying at the Centre de Recerca Matemática (Barcelona) in Spring 2006.


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repository.enu.kzrepository.enu.kz/bitstream/handle/123456789/1130/Convolution... · J Fourier Anal Appl (2011) 17:486–505 DOI 10.1007/s00041-010-9159-9 Convolution Inequalities