Nonlinear Differ. Equ. Appl. 18 (2011), 287–308c© 2011 Springer Basel AG1021-9722/11/030287-22published online January 13, 2011DOI 10.1007/s00030-010-0096-8

Nonlinear Differential Equationsand Applications NoDEA

Multilinear commutators of fractionalintegrals over Morrey spaces withnon-doubling measures

Xiangxing Tao and Taotao Zheng

Abstract. In this paper, the authors study the boundedness of multilin-ear fractional integrals on the product Morrey space with non-doublingmeasure, and investigate the Morrey boundedness properties of the multi-linear commutators generated by multilinear fractional integral operatorswith a tuple of RBMO functions.

Mathematics Subject Classification (2010). 42B20, 42B25, 42B30.

Keywords. Multi-commutator, Multilinear fractional integral,RBMO function, Morrey space, Non-doubling measure.

1. Introduction and main results

The classical result of commutator is due to Coifman et al. [1]: if b ∈ BMO andT is a Calderon–Zygmund operator, then the commutator [b, T ] is bounded onLp spaces for 1 < p < ∞. In 2001, Tolsa [2] developed the theory of Calderon-Zygmund operators and their commutators with RBMO functions in the set-ting of non-doubling measures. Later on, Chen and Sawyer [3] modified thedefinition of RBMO to investigate the commutators of the potential operatorsand RBMO functions. Recently, Sawano and Tanaka [4,5] defined the Morreyspaces for non-doubling measures, and obtained the boundedness of the com-mutators generated by fractional integrals with RBMO functions in Morreyspaces. Here we point out that the commutator estimates for non-doublingmeasures play an important role in solving the long-standing open questions,like the Painleve problem.

In recent years, the theory on multilinear integral operators and multilin-ear commutators have attracted much attentions as a rapid developing field in

This work was supported in part by the NNSF of China under Grant #10771110 and#10471069, and sponsored by the NSF of Ningbo City under Grant #2009A610090.

288 X. Tao and T. Zheng NoDEA

harmonic analysis. Hu et al. [6] studied the Lp-boundedness and certain weaktype endpoint estimates for multilinear commutators generated by multilinearCalderon-Zygmund singular integral with RBMO functions. In this paper,we will extend the works of Sawano and Tanaka [4,5,7] and Tao et al. [8] tothe context for multilinear commutators generated by multilinear fractionalintegrals and RBMO functions over the Morrey spaces with non-doublingmeasures.

Throughout this paper μ will be a positive Radon measures on Rd sat-

isfing the growth condition: there exist a constant C > 0 and n ∈ (0, d] suchthat

μ(Q) ≤ Cl(Q)n (1.1)

for any cube Q ∈ Rd with sides parallel to the coordinate axes. Q(x, l(Q)) will

be the cube centered at x with side length l(Q). For r > 0, rQ will denote thecube with the same center as Q and with l(rQ) = rl(Q). The set of all cubesQ ⊂ R

d satisfying μ(Q) > 0 is denoted by Q(μ).Before stating our main results, we fix some notations and define some

terminologies. Given βd ≥ 2d+1 large enough but depending only on the dimen-sion d, we say that a cube Q ⊂ R

d is doubling if μ(2Q) ≤ βdμ(Q). For any fixedcube Q ⊂ R

d, let N ≥ 0 be the smallest integer such that 2NQ is doubling.We denote this cube by ˜Q.

Let 0 ≤ α < n, for two cubes Q ⊂ R in Rd, we always set

K(α)Q,R = 1 +

NQ,R∑

k=1

[

μ(2kQ)l(2kQ)n

]1− αmn

,

where NQ,R is the first positive integer k such that l(2kQ) ≥ l(R). If α = 0,then K

(α)Q,R = KQ,R, The latter was introduced by Tolsa in [2].

Denote by mQf the mean value of f on Q, namely, mQf = 1μ(Q)

∫

Qf(x)dμ.

Let η > 1 be a fixed constant, we say that f ∈ L1loc(μ) is in RBMO(μ) if there

exist a constant A such that

1μ(ηQ)

∫

Q

|f(y) − m˜Qf |dμ(y) ≤ A (1.2)

for any cube Q, and

|mQf − mRf | ≤ AKQ,R (1.3)

for any two doubling cubes Q ⊂ R. The minimal constant A is the RBMO(μ)norm of f , and it will be denoted by ‖f‖∗.

We recall the definition of the Morrey space with non-doubling measure.

Definition 1.1. Let k > 1, and 1 ≤ q ≤ p < ∞, the Morrey space Mpq (k, μ) is

defined as

Mpq (k, μ) :=

{

f ∈ Lqloc(μ); ‖f‖Mp

q (k,μ) < ∞}

Vol. 18 (2011) Multilinear commutators with non-doubling measures 289

with the norm

‖f‖Mpq (k,μ) := sup

Q∈Q(μ)

μ(kQ)1p − 1

q

(∫

Q

|f |qdμ

)1q

.

As is easily seen, the space Mpq (k, μ) is a Banach space with its norm.

The Morrey space norm reflects local regularity of f more precisely than theLebesgue space norm. It is easy to see from the Holder inequality that Lp(μ) =Mp

p (k, μ) ⊂ Mpq1

(k, μ) ⊂ Mpq2

(k, μ) whenever 1 ≤ q2 ≤ q1 ≤ p < ∞. Moreover,the definition of the spaces is independent of the constant k > 1, and thenorms for different choice of k > 1 are equivalent, see [4,5,7] for details. Wewill denote Mp

q (2, μ) by Mpq (μ).

Denoting by �f = (f1, f2, . . . , fm), we consider in this article the multilin-ear fractional integral operator Iα,m as follows,

Iα,m(�f)(x) =∫

(Rd)m

f1(y1) · · · fm(ym)dμ(y1) · · · dμ(ym)|(x − y1, x − y2, . . . , x − ym)|mn−α

for x ∈ Rd. (1.4)

Let bi ∈ RBMO(μ) for i = 1, 2, . . . ,m, and let �b = (b1, b2, . . . , bm), thenthe multilinear commutator [�b, Iα,m] is formally defined as[

�b, Iα,m

]

(�f)(x) =∫

(Rd)m

m∏

j=1

(bj(x) − bj(yj))f1(y1) · · · fm(ym)dμ(y1) · · · dμ(ym)|(x − y1, x − y2, . . . , x − ym)|mn−α

.

More generally, denote by Cmi the family of all subsets σ = {σ1, σ2, . . . , σi}

of i different elements of {1, 2, . . . ,m}, and let σ′ = {1, 2, . . . ,m}\σ and−→bσ =

{bσ1 , bσ2 , . . . , bσi}. For any σ ∈ Cm

i , we define[−→bσ , Iα,m

]

(�f)(x) =∫

(Rd)m

∏

σj∈σ

(bσj(x) − bσj

(yj))

×f1(y1) · · · fm(ym)dμ(y1) · · · dμ(ym)|(x − y1, x − y2, . . . , x − ym)|mn−α

.

In case σ = {1, 2, . . . ,m}, one sees that [−→bσ , Iα,m] is just the commutator

[−→b , Iα,m].

Here we remark that the multilinear commutator [�b, Iα,m] can be equiv-alently rewritten as

[

�b, Iα,m

]

(�f)(x) = [bm, [bm−1, . . . , [b1, Iα,m

m︷ ︸︸ ︷

] · · · ]](�f)(x). (1.5)

The main results in this paper can be stated as follows.

Theorem 1.2. Suppose that 0 < α < mn and 1 < qi ≤ pi < ∞, and let1p = 1

p1+ 1

p2+ · · · + 1

pm− α

n > 0, 1q = 1

q1+ 1

q2+ · · · + 1

qm− α

n > 0, then

‖Iα,m(�f)‖Mpq (μ) ≤ C

m∏

i=1

‖fi‖Mpiqi (μ)

with the constant C independent of �f .

290 X. Tao and T. Zheng NoDEA

Remark 1.3. In the case pi = qi, Theorem 1.2 implies the Lp-boundedness forthe operator Iα,m, precisely,

‖Iα,m(�f)‖Lp(μ) ≤ Cm∏

i=1

‖fi‖Lpi (μ)

with 1p = 1

p1+ 1

p2+ · · · + 1

pm− α

n > 0 for any 1 < pi < ∞.

Theorem 1.4. Let 0 < α < mn and 1 < qi ≤ pi < ∞, and let 1p = 1

p1+ 1

p2+

· · · + 1pm

− αn > 0, 1

q = 1q1

+ 1q2

+ · · · + 1qm

− αn > 0. Suppose ‖μ‖=∞, and

that bi ∈ RBMO(μ) for i = 1, 2, . . . , m. Then, for all σ ⊆ {1, 2, . . . ,m}, thecommutators [�bσ, Iα,m] are bounded from Mp1

q1(μ) × Mp2

q2(μ) × · · · × Mp1

q1(μ) to

Mpq (μ), moreover,

∥

∥

∥

[

�bσ, Iα,m

]

(�f)∥

∥

∥

Mpq (μ)

≤ C∏

j∈σ

‖bj‖∗m∏

i=1

‖fi‖Mpiqi (μ),

and particularly,∥

∥

∥

[

�b, Iα,m

]

(�f)∥

∥

∥

Mpq (μ)

≤ C

m∏

i=1

‖bi‖∗‖fi‖Mpiqi (μ),

where C > 0 is the constant independent of �f , �b and �bσ.

Remark 1.5. In the case m = 1, Theorem 1.2 and 1.4 follow that the fractionalintegral operator Iα and its commutator [b, Iα] are all bounded from Mp1

q1(μ)

to Mpq (μ), which are just Theorem 3.3 of [4] and Theorem 4.6 of [5], so our

theorems contain their conclusions. In [8] the authors obtained the same resultas Theorem 1.2 if μ is the Lebesgue measure.

To show Theorem 1.2 in Sect. 2, we will use a maximal function Mkf(x)and give a pointwise estimate for the multilinear fractional integral Iα,m(�f)(x)in the setting of non-doubling measures. In order to derive Theorem 1.4, we willexploit the decomposition technique for the product of functions and overcomethe difficult brought from the multilinear operators and non-doubling mea-sures. We will establish a pointwise estimate for the sharp maximal functionof the commutator [�b, Iα,m](�f) by using the non-centered maximal operatorM

(α)τ,η , see Lemma 3.4 below. We will prove Theorem 1.4 in Sect. 3.

2. The proof of Theorem 1.2

Before proving the theorem, we recall the following maximal operator,

Mkf(x) = supx∈Q⊂Q(μ)

1μ(kQ)

∫

Q

|f(y)|dμ(y),

and some lemmas which will be used in this paper.

Vol. 18 (2011) Multilinear commutators with non-doubling measures 291

Lemma 2.1. [4] If 1 < q ≤ p < ∞, then there exist a constant C independentof f such that

‖Mkf‖Mpq (μ) ≤ C‖f‖Mp

q (μ).

Lemma 2.2. For 0 < α < mn, 1 < qi ≤ pi < ∞, 1h = 1

p1+ 1

p2+ · · · + 1

pm,

1p = 1

h − αn > 0, we have

|Iα,m�f(x)| ≤ C

(

m∏

i=1

‖fi‖Mpiqi (μ)

)1− hp(

m∏

i=1

Mkfi(x)

)hp

with the absolute constant C independent of �f .

Proof. For x, yi ∈ Rd and x = yi, i = 1, 2, . . . ,m, we can see from the Fubini

theorem that

Iα,m(�f)(x) ≤∫

(Rd)m

|f1(y1)||f2(y2)| · · · |fm(ym)|dμ(y1)dμ(y2) · · · dμ(ym)|(x − y1, x − y2, . . . , x − ym)|mn−α

≤ C

∫

(Rd)m

∫ ∞

|(x−y1,x−y2,...,x−ym)|lα−mn−1dl

(

m∏

i=1

|fi(yi)|dμ(yi)

)

≤ C

∫ ∞

0

(

m∏

i=1

1ln

∫

|(x−yi)|<l

|fi(yi)|dμ(yi)

)

lα−1dl. (2.1)

Using the Holder inequality and the growth condition (1.1) for the non-doubling measure μ, and letting cube Q = Q(x, 2l), then we have

∫ λ

0

(

m∏

i=1

1ln

∫

|(x−yi)|<l

|fi(yi)|dμ(yi)

)

lα−1dl

≤ C

∫ λ

0

(

m∏

i=1

1μ(kQ)

∫

Q

|fi(yi)|dμ(yi)

)

lα−1dl

≤ C

∫ λ

0

(

m∏

i=1

Mkfi(x)

)

lα−1dl ≤ Cλαm∏

i=1

Mkfi(x).

On the other hand, with the similar ways as above we have

1ln

∫

|(x−yi)|<l

|fi(yi)|dμ(yi) ≤ C

ln

∫

Q

|fi(yi)|dμ(yi)

≤ C

ln

(∫

Q

|fi(yi)|qidμ(yi))

1qi

μ(Q)1− 1qi

≤ Cl− n

pi ‖fi‖Mpiqi (μ)

292 X. Tao and T. Zheng NoDEA

which means that∫ ∞

λ

(

m∏

i=1

1ln

∫

|(x−yi)|<l

|fi(yi)|dμ(yi)

)

lα−1dl

≤ C

∫ ∞

λ

(

m∏

i=1

l− n

pi ‖fi‖Mpiqi (μ)

)

lα−1dl ≤ Cλ− np

m∏

i=1

‖fi‖Mpiqi (μ).

Thus, by (2.1), we obtain that

|Iα,m�f(x)| ≤ C

[

λαm∏

i=1

Mkfi(x) + λ− np

m∏

i=1

‖fi‖Mpiqi (μ)

]

.

This yields the lemma by choosing the suitable positive real number λ. �

Now we give the proof of theorem 1.2. Note 0 < α < mn, 1 < qi ≤ pi <∞, 1

h = 1p1

+ 1p2

+ · · · + 1pm

, and let 1r = 1

q1+ 1

q2+ · · · + 1

qm. By the lemma 2.2,

one can get that

‖Iα,m�f‖Mp

q (μ) ≤ C

(

m∏

i=1

‖fi‖Mpiqi (μ)

)1− hp

× supQ∈Q(μ)

|μ(kQ)| 1p − 1

q

⎧

⎨

⎩

∫

Q

(

m∏

i=1

Mkfi(y)

)qhp

dμ(y)

⎫

⎬

⎭

1q

.

Since prqh = n−rα

n−hα > 1, hence by the Holder inequality and Lemma 2.1 it follows

|μ(kQ)| 1p − 1

q

⎧

⎨

⎩

∫

Q

(

m∏

i=1

Mkfi(y)

)qhp

dμ(y)

⎫

⎬

⎭

1q

≤ C|μ(kQ)| 1p − h

pr

{

∫

Q

(

m∏

i=1

Mkfi(y)

)r

dμ(y)

}hpr

≤ C

⎧

⎨

⎩

|μ(kQ)| 1h − 1

r

{

∫

Q

(

m∏

i=1

Mkfi(y)

)r

dμ(y)

}1r

⎫

⎬

⎭

hp

≤ C

{

m∏

i=1

|μ(kQ)| 1pi

− 1qi

(∫

Q

(Mkfi(y))qi dμ(y))

1qi

}hp

≤ C

(

m∏

i=1

‖Mkfi‖Mpiqi (μ)

)hp

≤ C

(

m∏

i=1

‖fi‖Mpiqi (μ)

)hp

.

Thus the theorem is obtained.

Vol. 18 (2011) Multilinear commutators with non-doubling measures 293

3. The proof of Theorem 1.4

In this section we will use the sharp maximal estimates. Let f be a functionin L1

loc(μ), the sharp maximal function of f is defined by

M �,(α)f(x) = supQ�x

1μ(

32Q

)

∫

Q

|f(y) − m˜Qf |dμ(y)

+ supR⊃Q�xQ,R doubling

|mQf − mRf |K

(α)Q,R

. (3.1)

The non-centered doubling maximal operator is defined by

Nf(x) = supQ�x

Q doubling

1μ(Q)

∫

Q

|f(y)|dμ(y). (3.2)

By the Lebesgue differential theorem, it is easy to see that |f(x)| ≤ Nf(x) forany f ∈ L1

loc(μ) and μ-a.e. x ∈ Rd. Define the non-centered maximal operator,

M (α)τ,η f(x) = sup

Q�x

(

1μ(ηQ)1− τα

n

∫

Q

|f(y)|τdμ(y))

1τ

(3.3)

for η > 1 and τ > 1, where the supremum is taking over all the cubes Qcontaining the point x. Usually we denote M �,(0)(f) simply by M �(f) andM

(0)τ,η (f) by Mτ,η(f).

From the work in [2], we know that Mτ,η is bounded on Lq(μ) for q >τ > 1 and η > 1. Moreover, we have the following lemma.

Lemma 3.1. Let q > τ > 1, η > 1, and 1 < q ≤ p < ∞, then the operator Mτ,η

is bounded on Mpq (μ) and

‖Mτ,η(f)‖Mpq (μ) ≤ C ‖f‖Mp

q (μ)

with the constant C independent of f .

Proof. The lemma could be proved along the similar lines as that ofTheorem 2.3 in [4], i.e. Lemma 2.1 of this paper. Fix a cube Q0 ∈ Q(μ)and put L = l(Q0)

2 , and let f1 = χ η+7η−1 Q0

f and f2 = f − f1, then we have forall y ∈ Q0 that

Mτ,ηf(y) ≤ Mτ,ηf1(y) + Mτ,ηf2(y). (3.4)

Noting for y ∈ Q0,

Mτ,ηf2(y) ≤ supy∈Q∈Q(μ)l(Q)≥ 8L

η−1

{

1μ(ηQ)

∫

Q

|f |τdμ

}1τ

≤ supQ0⊂Q∈Q(μ)

⎧

⎨

⎩

1

μ(

2η1+η Q

)

∫

Q

|f |τdμ

⎫

⎬

⎭

1τ

(3.5)

294 X. Tao and T. Zheng NoDEA

because of the fact that y ∈ Q0

⋂

Q and l(Q) ≥ 8L(η−1) yields Q0 ⊂ 1+η

2 Q.Thus, from the inequalities (3.4) and (3.5), the Lq(μ)-boundedness for Mτ,η,and the Holder inequality we get that

μ

(

2η(η + 7)η2 − 1

Q0

)1p − 1

q(∫

Q0

(Mτ,ηf)qdμ

)1q

≤ μ

(

2η(η + 7)η2 − 1

Q0

)1p − 1

q(∫

Q0

(Mτ,ηf1)qdμ

)1q

+μ (Q0)1p − 1

q

(∫

Q0

(Mτ,ηf2)qdμ

)1q

≤ μ

(

2η(η + 7)η2 − 1

Q0

)1p − 1

q

(

∫

η+7η−1 Q0

|f |qdμ

)1q

+ supQ0⊂Q∈Q(μ)

μ(Q0)1p

μ(

2η1+η Q

)1τ

(∫

Q

|f |τdμ

)1τ

≤ C supQ0⊂Q∈Q(μ)

μ

(

2η

1 + ηQ

)1p − 1

q(∫

Q

|f |qdμ

)1q

which implies the lemma. �

Lemma 3.2. Let 1 < τ < nα , 1 < τ < q ≤ p < ∞, 1 < t ≤ s < ∞, 1

s = 1p − α

n ,qp = t

s , then

‖M (α)τ,η f‖Ms

t (μ) ≤ C‖f‖Mpq (μ)

with the constant C independent of f .

Proof. Noting the condition qp = t

s , it is enough to show the following pointwiseestimates

M (α)τ,η (f)(x) ≤ C

(‖f‖Mpq (μ)

)1− ps (Mτ,ηf(x))

ps . (3.6)

To do this, we fixed the point x such that M(α)τ,η (f)(x) > 0. We choose βx =

(‖f‖M

pq (μ)

Mτ,ηf(x) )p and then write

M (α)τ,η f(x) ≤ sup

x∈Q∈Q(μ)μ(ηQ)≤βx

(

1μ(ηQ)1− τα

n

∫

Q

|f(y)|τdμ(y))

1τ

+ supx∈Q∈Q(μ)μ(ηQ)>βx

(

1μ(ηQ)1− τα

n

∫

Q

|f(y)|τdμ(y))

1τ

≤ (βx)αn sup

x∈Q∈Q(μ)μ(ηQ)≤βx

(

1μ(ηQ)

∫

Q

|f(y)|τdμ(y))

1τ

Vol. 18 (2011) Multilinear commutators with non-doubling measures 295

+ supx∈Q∈Q(μ)μ(ηQ)>βx

μ(ηQ)−( 1p − α

n )‖f‖Mpq (μ)

by using the Holder inequality. Hence we have

M (α)τ,η f(x) ≤ (βx)

αn Mτ,ηf(x) + (βx)− 1

s ‖f‖Mpq (μ),

which yields the desired inequality (3.6). The lemma is proved. �

Lemma 3.3. [5] Suppose that 1 < q ≤ p < ∞, and there exists an increasingsequence of concentric doubling cubes, I0 ⊂ I1 ⊂ · · · ⊂ Ik ⊂ · · ·, such that

limk→∞

mIk(f) = 0, and

∞⋃

k=0

Ik = Rd.

Then there exist a constant C > 0 independent on f such that

‖Nf‖Mpq (μ) ≤ C‖M �,(α)f‖Mp

q (μ).

In order to investigate the multilinear commutators [�b, Iα,m] and [�bσ,Iα,m],we introduce the following decomposition technique. For any

−→λ = (λ1, λ2, . . . ,

λm) ∈ Rm, writing

−→bσ(x)−−→

bσ(y) = (−→bσ(x)−−→

λσ)− (−→bσ(y)−−→

λσ), thus it is clearthat [

−→bσ , Iα,m](�f) = [(

−→bσ − −→

λσ), Iα,m](�f). Moreover,

[−→b , Iα,m

]

(�f)(x) =m∑

i=0

∑

σ∈Cmi

(−1)m−i∏

σj∈σ

(bσj(x) − λσj

)

×∫

(Rd)m

∏

σk∈σ′(bσk

(y) − λσk)

× f1(y1) · · · fm(ym)dμ(y1) · · · dμ(ym)|(x − y1, x − y2, . . . , x − ym)|mn−α

. (3.7)

By expanding bσk(y) − λσk

= (bσk(y) − bσk

(x)) + (bσk(x) − λσk

) and∏

σk∈σ′(bσk

(y)− λσk)=

∑

σ(1)∪σ(2)=σ′σ(1)∩σ(2)=∅

∏

σu∈σ(1)

(bσu(y) − bσu

(x))∏

σv∈σ(2)

(bσv(x)− λσv

),

hence we can obtain from the equality (3.7) that[−→

b , Iα,m

]

(�f)(x) = (−1)m+1m∏

j=1

(bj(x) − λj)Iα,m(�f)(x)

+ (−1)mIα,m

(

(�b − �λ)�f)

(x)

+m−1∑

i=1

∑

σ∈Cmi

Cm,i

∏

σj∈σ

(bσj(x) − λσj

)

×[−→bσ′ , Iα,m

]

(�f)(x), (3.8)

where Cm,i are constants depending only on m and i.

296 X. Tao and T. Zheng NoDEA

Remember these notations and the equality (3.8) in mind, we turn toprove the following lemma, which is crucial to us.

Lemma 3.4. Let 0 < α < mn, τ > 1, si > 1, bi ∈ RBMO(μ) and fi ∈ Lqi(μ),i = 1, 2, . . . ,m, then there exist a constant C > 0 independent of bi and fi

such that

M �,(α)([

�b, Iα,m

]

(�f))

(x) ≤ C

[

‖b1‖∗ . . . ‖bm‖∗Mτ, 32(Iα,m(�f))(x)

+m−1∑

i=1

∑

σ∈Cmi

∏

j∈σ

‖bj‖∗Mτ, 32

×([−→

bσ′ , Iα,m

]

(�f))

(x)

+m∏

i=1

‖bi‖∗M( α

m )si,

98

fi(x)

]

. (3.9)

Proof. For simplicity, we denote by R(�b, �f)(x) the quantities on the right handside of the inequality (3.9). Recall the definition of the sharp maximal operatorM �,(α), and use the standard technique, see [3] for example, we only need toprove that, for any x ∈ R

d and a cube Q � x, there is a quantity hQ such that

1μ(

32Q

)

∫

Q

∣

∣

∣[�b, Iα,m](�f)(z) − hQ

∣

∣

∣ dμ(z) ≤ CR(�b, �f)(x) (3.10)

and

|hQ − hR| ≤ C(

KQ,RK(α)Q,R

)m

R(�b, �f)(x) (3.11)

with the absolute constant C independent of �b, �f,Q and R, where R is anydoubling cube with Q ⊂ R. In fact, we take

hQ = (−1)mmQ

(

Iα,m

(

(m˜Q(b1) − b1)f1χRd\ 4

3 Q, . . . , (m˜Q(bm)

−bm)fmχRd\ 4

3 Q

))

,

and clearly

hR = (−1)mmR

(

Iα,m

(

(mR(b1) − b1)f1χRd\ 43 R, . . . , (mR(bm)

−bm)fmχRd\ 4

3 R

))

.

Vol. 18 (2011) Multilinear commutators with non-doubling measures 297

Recall the equality (3.8), for any z ∈ Q, we have that

∣

∣

∣

[−→b , Iα,m

]

(�f)(z) − hQ

∣

∣

∣ ≤∣

∣

∣

∣

∣

∣

m∏

j=1

(bj(x) − m˜Q(bj))Iα,m(�f)(z)

∣

∣

∣

∣

∣

∣

+

∣

∣

∣

∣

∣

∣

m−1∑

i=1

∑

σ∈Cmi

Cm,i

∏

σj∈σ

(bσj(z) − m

˜Q(bσj))

×[−→bσ′ , Iα,m

]

(�f)(z)∣

∣

∣

+∣

∣

∣(−1)mIα,m

(

(�b − m˜Q(�b))�f

)

(z) − hQ

∣

∣

∣

=: I(z) + II(z) + III(z). (3.12)

In order to show the inequality (3.10), we will calculate the integralsfor the three functions above, respectively. Firstly, for τ > 1, by the Holderinequality one sees that

1μ(

32Q

)

∫

Q

|I(z)|dμ(z)

≤m∏

i=1

(

1μ(

32Q

)

∫

Q

|(bi(z) − m˜Q(bi))|τidμ(z)

)1τi

×(

1μ(

32Q

)

∫

Q

|Iα,m(�f)|τdμ(z)

)1τ

≤ C‖b1‖∗‖b2‖∗ . . . ‖bm‖∗Mτ, 32(Iα,m(�f))(x), (3.13)

where we have choose τi > 1 such that 1τ1

+ 1τ2

+ · · · + 1τm

+ 1τ = 1.

Similarly, for τ > 1, by the Holder inequality, we also deduce that

1μ(

32Q

)

∫

Q

|II(z)|dμ(z)

≤ C

m−1∑

i=1

∑

σ∈Cmi

∏

σj∈σ

‖bσj‖∗Mτ, 3

2

([−→bσ′ , Iα,m

]

(�f))

(x). (3.14)

To estimate the integral related to the function III(z), we split each fi

as fi = f0i + f∞

i , where f0i = fχ 4

3 Q and f∞i = fi − f0

i , this yields

m∏

j=1

fj(yj) =∑

α1,α2,...,αm∈{0,∞}fα11 (y1) . . . fαm

m (ym)

=m∏

j=1

f0j (yj) +

∑

∗fα11 (y1) . . . fαm

m (ym) +m∏

j=1

f∞j (yj), (3.15)

where each term in∑

∗ satisfies that αj1 = αj2 = · · · = αjλ= 0, for some

1 ≤ λ < m and some {j1, j2, . . . , jλ} ⊂ {1, 2, . . . ,m}. So we can decompose

298 X. Tao and T. Zheng NoDEA

the function III(z) further into three parts as follows

III(z) ≤∣

∣

∣Iα,m

(

(b1 − m˜Q(b1))f0

1 , . . . , (bm − m˜Q(bm))f0

m

)

(z)∣

∣

∣

+∑

∗

∣

∣

∣Iα,m

(

(b1 − m˜Q(b1))fα1

1 , . . . , (bm − m˜Q(bm))fαm

m

)

(z)∣

∣

∣

+∣

∣

∣(−1)mIα,m

(

(b1 − m˜Q(b1))f∞

1 , . . . , (bm

−m˜Q(bm))f∞

m

)

(z) − hQ

∣

∣

∣

=: III1(z) + III2(z) + III3(z). (3.16)

For si > 1, we can take 1 < ui < si such that 1v = 1

u1+ 1

u2+ · · · +

1um

− αn > 0 and v > 1. Let 1

ui= 1

si+ 1

tifor each i = 1, 2, . . . ,m, then

1 < ti < ∞. Using Theorem 1.2, we know that the operator Iα,m is boundedfrom Lu1(μ) × Lu2(μ) × · · · × Lum(μ) to Lv(μ). Hence, by this boundednessand the Holder inequality, we have

1μ(

32Q

)

∫

Q

|III1(z)|dμ(z)

≤ μ(Q)1− 1v

μ(

32Q

) ‖Iα,m

(

(b1 − m˜Q(b1))f0

1 , . . . , (bm − m˜Q(bm))f0

m

)

(z)‖Lv(μ)

≤ Cμ

(

32Q

)− 1v

m∏

i=1

‖(bi − m˜Q(bi))f0

i ‖Lui (μ)

≤ Cμ

(

32Q

)− 1v

m∏

i=1

(

∫

( 43 Q)

|fi|sidμ(yi)

)1si

×(

∫

( 43 Q)

|bi − m˜Q(bi)|tidμ(yi)

)1ti

≤ C

m∏

i=1

⎛

⎝

1

μ(

32Q

)1− αsimn

∫

( 43 Q)

|fi|sidμ

⎞

⎠

1si

×(

1μ(

32Q

)

∫

( 43 Q)

|bi − m˜Q(bi)|tidμ

)1ti

≤ C

m∏

i=1

‖bi‖∗M( α

m )si,

98

fi(x). (3.17)

In order to estimate the integral of terms III2(z) and III3(z) over Q, wewill give their point-wise estimates. In fact, for z ∈ Q, since 1 ≤ λ ≤ m − 1 weobserve that

Vol. 18 (2011) Multilinear commutators with non-doubling measures 299

III2(z) =∑

∗

∣

∣

∣Iα,m

(

(b1 − m˜Q(b1))fα1

1 , . . . , (bm − m˜Q(bm))fαm

m

)

(z)∣

∣

∣

≤ C∑

∗

∏

j∈{j1,...,jλ}

∫

43 Q

|bj(yj) − m˜Q(bj)||fj(yj)|dμ(yj)

×∫

(Rd\ 43 Q)m−λ

∏

j /∈{j1,...,jλ} |bj(yj) − m˜Q(bj)||fj(yj)|dμ(yj)

(

∑

j /∈{j1,...,jλ} |z − yj |)mn−α

≤ C∑

∗μ

(

32Q

)λ− αλmn ∏

j∈{j1,...,jλ}

×⎛

⎝

1

μ(

32Q

)1− αsjmn

∫

43 Q

|fj(yj)|sj dμ(yj)

⎞

⎠

1sj

×(

1μ(

32Q

)

∫

43 Q

|bj(yj) − m˜Q(bj)|sj

′dμ(yj)

)1

sj′

×∞∑

k=1

∫

(2k 43 Q)m−λ

∏

j /∈{j1,...,jλ} |bj(yj) − m˜Q(bj)||fj(yj)|dμ(yj)

l(

2k 43Q

)mn−α

≤ C∑

∗

⎛

⎝

∏

j∈{j1,...,jλ}‖bj‖∗M

( αm )

sj , 98

fj(x)

⎞

⎠

∞∑

k=1

2−(k−1)(n− αm )λ

×∫

(2k 43 Q)m−λ

∏

j /∈{j1,...,jλ} |bj(yj) − m˜Q(bj)||fj(yj)|dμ(yj)

l(

2k 43Q

)(n− αm )(m−λ)

≤ C∑

∗

⎛

⎝

∏

j∈{j1,...,jλ}‖bj‖∗M

( αm )

sj , 98

fj(x)

⎞

⎠

∞∑

k=1

2−(k−1)(n− αm )λ

×∏

j /∈{j1,...,jλ}

1

l(

2k 43Q

)(n− αm )

∫

(2k 43 Q)

|bj(yj) − m˜Q(bj)||fj(yj)|dμ(yj)

≤ C∑

∗

⎛

⎝

∏

j∈{j1,...,jλ}‖bj‖∗M

( αm )

sj , 98

fj(x)

⎞

⎠

∞∑

k=1

2−(k−1)(n− αm )λ

×∏

j /∈{j1,...,jλ}

⎡

⎣

(

1l(

2k 43Q

)n

∫

(2k 43 Q)

|bj(yj) − m˜Q(bj)|s′

j dμ(yj)

)1

s′j

×⎛

⎝

1

l(

2k 43Q

)(n− αsjm )

∫

(2k 43 Q)

|fj(yj)|sj dμ(yj)

⎞

⎠

1sj

⎤

⎥

⎦

300 X. Tao and T. Zheng NoDEA

≤ C∑

∗

∏

j∈{j1,...,jλ}‖bj‖∗M

( αm )

sj , 98

fj(x)∞∑

k=1

2−k(n− αm )λ

×∏

j /∈{j1,...,jλ}‖bj‖∗(k + 1)M( α

m )sj , 9

8fj(x)

≤ C

m∏

j=1

‖bj‖∗M( α

m )sj , 9

8fj(x),

where we have used the fact (see [2]) that, there is an absolute constant Csuch that, for any b ∈ RBMO, integer k ≥ 0 and cubes Q,

|m˜2k 4

3 Q(b) − m

˜Q(b)| ≤ C‖b‖∗K˜Q,˜2k 4

3 Q≤ C‖b‖∗KQ,2k 4

3 Q ≤ Ck‖b‖∗.(3.18)

On the other hand, for III3(z), we note for any z, y ∈ Q that∣

∣

∣Iα,m

(

(b1 − m˜Q(b1))f∞

1 , . . . , (bm − m˜Q(bm))f∞

m

)

(z)

− Iα,m

(

(b1 − m˜Q(b1))f∞

1 , . . . , (bm − m˜Q(bm))f∞

m

)

(y)∣

∣

∣

≤∫

(Rd\ 43 Q)m

∣

∣

∣

∣

1|(z − y1, . . . , z − ym)|mn−α

− 1|(y − y1, . . . , y − ym)|mn−α

∣

∣

∣

∣

×∣

∣

∣

∣

∣

m∏

i=1

(bi(yi) − m˜Q(bi))f∞

i

∣

∣

∣

∣

∣

dμ(y1)dμ(y2) · · · dμ(ym)

≤ C

∫

(Rd\ 43 Q)m

|z − y|∏mi=1 |(bi(yi) − m

˜Q(bi))f∞i |

|(y − y1, . . . , y − ym)|mn−α+1

×dμ(y1)dμ(y2) · · · dμ(ym)

≤ Cm∏

i=1

∞∑

k=1

∫

2k 43 Q\2k−1 4

3 Q

l(Q)1m

l(

2k 32Q

)n− αm + 1

m

|(bi(yi) − m˜Q(bi))fi|dμ(yi)

≤ C

m∏

i=1

∞∑

k=1

2−k/m

(

1l(2k 3

2Q)n

∫

2k 43 Q

|(bi(yi) − m˜Q(bi))|si

′dμ(yi)

)1

si′

×(

1l(2k 3

2Q)n−(αsi/m)

∫

2k 43 Q

|fi(yi)|sidμ(yi)

)1si

≤ C

m∏

i=1

∞∑

k=1

2−k/m(k + 1)‖bi‖∗M( α

m )si,

98

fi(x) ≤ C

m∏

i=1

‖bi‖∗M( α

m )si,

98

fi(x),

where we have used the fact (3.18) again.Taking the mean over y ∈ Q, we can obtain that

1μ(

32Q

)

∫

Q

(|III2(z)| + |III3(z)|) dμ(z) ≤ C

m∏

i=1

‖bi‖∗M( α

m )si,

98

fi(x). (3.19)

Vol. 18 (2011) Multilinear commutators with non-doubling measures 301

Combing the inequalities (3.12), (3.13), (3.14), (3.16), (3.17) and (3.19),we see from the estimates of I, II, III1, III2 and III3 that the desired inequal-ity (3.10) holds.

Next we turn to estimate the inequality (3.11). For any cubes Q ⊂ Rwith x ∈ Q, where R is doubling. We denote NQ,R + 2 by N , then 2NQ ⊃ 2Qand 2NQ ⊃ 2R. We recall the equality (3.15) and let f0

i = fiχ2N Q\ 43 Q and

fRi = fiχ2N Q\ 4

3 R, and let f∞i = fiχRd\2N Q. Then we can write

|hQ − hR|=

∣

∣

∣mQ

[

Iα,m

(

(m˜Q(b1) − b1)f1χRd\ 4

3 Q, . . . , (m˜Q(bm) − bm)fmχ

Rd\ 43 Q

)]

− mR

[

Iα,m

(

(mR(b1) − b1)f1χRd\ 43 R, . . . , (mR(bm) − bm)fmχ

Rd\ 43 R

)]∣

∣

∣

≤∣

∣

∣mQ

[

Iα,m

(

(m˜Q(b1) − b1)f∞

1 , . . . , (m˜Q(bm) − bm)f∞

m

)]

− mR

[

Iα,m

(

(m˜Q(b1) − b1)f∞

1 , . . . , (m˜Q(bm) − bm)f∞

m

)]∣

∣

∣

+∣

∣

∣mR

[

Iα,m

(

(m˜Q(b1) − b1)f∞

1 , . . . , (m˜Q(bm) − bm)f∞

m

)]

− mR [Iα,m ((mR(b1) − b1)f∞1 , . . . , (mR(bm) − bm)f∞

m )]|+

∑

α1,α2,...,αm∈{0,∞}at least one αi =∞

∣

∣

∣mQ

[

Iα,m

(

(m˜Q(b1) − b1)fα1

1 , . . . ,

× (m˜Q(bm) − bm)fαm

m

)]∣

∣

∣

+∑

α1,α2,...,αm∈{R,∞}at least one αi =∞

|mR [Iα,m ((mR(b1) − b1)fα11 , . . . ,

× (mR(bm) − bm)fαmm )]|

= A1 + A2 + A3 + A4.

For the term A1, noting

|mR(bi) − m˜Q(bi)| ≤ C‖bi‖∗KQ,R

and the similar argument as that for the estimate of III3, we can obtain that

A1 ≤ C

m∏

i=1

KQ,R‖bi‖∗M( α

m )si,

98

fi(x) ≤ C(KQ,R)mm∏

i=1

‖bi‖∗M( α

m )si,

98

fi(x).

(3.20)

To estimate A2, we recall the notations and note that, for any sequencesξj and ζj ,

m∏

j=1

(ξj + ζj) =m∑

i=0

∑

σ∈Cmi

∏

j∈σ

ξj

∏

j′∈σ′ζj′ .

302 X. Tao and T. Zheng NoDEA

Using this equality and expanding m˜Q(bj)−bj(y) = (m

˜Q(bj)−bj(z))+(bj(z)−bj(y)), we observe that

Iα,m

(

(m˜Q(b1) − b1)f∞

1 , . . . , (m˜Q(bm) − bm)f∞

m

)

(z)

=m∑

i=0

∑

σ∈Cmi

∏

j∈σ

(m˜Q(bj) − bj(z))

[

�bσ′ , Iα,m

]

(�fχRd\2N Q)(z).

Similarly,

Iα,m ((mR(b1) − b1)f∞1 , . . . , (mR(bm) − bm)f∞

m ) (z)

=m∑

i=0

∑

σ∈Cmi

∏

j∈σ

(mR(bj) − bj(z))[

�bσ′ , Iα,m

]

(�fχRd\2N Q)(z).

Thus∣

∣

∣Iα,m

(

(m˜Q(b1) − b1)f∞

1 , . . . , (m˜Q(bm) − bm)f∞

m

)

(z)

− Iα,m ((mR(b1) − b1)f∞1 , . . . , (mR(bm) − bm)f∞

m ) (z)|

≤∣

∣

∣

∣

∣

∣

m∑

i=1

∑

σ∈Cmi

∏

j∈σ

(m˜Q(bj) − bj(z))

[

�bσ′ , Iα,m

]

(�fχRd\2N Q)(z)

∣

∣

∣

∣

∣

∣

+

∣

∣

∣

∣

∣

∣

m∑

i=1

∑

σ∈Cmi

∏

j∈σ

(mR(bj) − bj(z))[

�bσ′ , Iα,m

]

(�fχRd\2N Q)(z)

∣

∣

∣

∣

∣

∣

=: B1(z) + B2(z). (3.21)

To estimate the integrals above, we recall that f∞j = fjχRd\2N Q and let

fQj = fjχ2N Q, then we can write that f∞

j = fj − fQj and fj = f∞

j + fQj , and

thus we havem∏

j=1

f∞j (yj) =

m∏

j=1

fj(yj) +m∑

i=1

(−1)i∑

�∈Cmi

∏

j∈�

fQj (yj)

∏

j′∈�′fj′(yj′)

=m∏

j=1

fj(yj) +m∑

λ=1

∑

{j1,j2,...,jλ}⊂{1,2,...,m}

×Cj1,j2,...,jmfQ

j1(yj1) . . . fQ

jλ(yjλ

)f∞jλ+1

(yjλ+1) . . . f∞jm

(yjm)

(3.22)

where Cj1,j2,...,jmare constants independent of �f and Q. From the equality

(3.22), we can deduce that[

�bσ′ , Iα,m

]

(�fχRd\2N Q)(x) =[

�bσ′ , Iα,m

]

(�f)(z)

+m∑

λ=1

∑

{j1,j2,...,jλ}⊂{1,2,...,m}Cj1,j2,...,jλ

Iα,m(g1F1, g2F2, . . . , gmFm)(z)

(3.23)

Vol. 18 (2011) Multilinear commutators with non-doubling measures 303

where

gj(y) ={

bj(z) − bj(y), if j ∈ σ′

1, if j ∈ σ

and

Fj(y) ={

fQj (y), if j ∈ {j1, j2, . . . , jλ}

f∞j (y), if j ∈ {j1, j2, . . . , jλ} .

Along the same lines as that of the pointwise estimates of III2(z), we canobtain that, for x, z ∈ R ⊂ 2N−1Q and if 1 ≤ λ ≤ m,

|Iα,m(g1F1, g2F2, . . . , gmFm)(z)|≤ C

∏

j∈{j1,j2,...,jλ}

1l(2NQ)n− α

m

∫

2N Q

|gj(yj)fj(yj)|dμ(yj)

×∞∑

k=1

2−k(n− αm )λ

∏

j ∈{j1,j2,...,jλ}1≤j≤m

1l(2k2NQ)n− α

m

×∫

2k2N Q

|gj(yj)fj(yj)|dμ(yj). (3.24)

Let (gj)∗k,s′

j= 1 if j ∈ σ; and

(gj)∗k,s′

j=

⎧

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎩

(

1l(2N Q)n

∫

2N Q|bj(z) − bj(y)|s′

j dμ(y))

1s′

j , if j ∈ σ′∩{j1, j2, . . . , jλ}

(

1l(2k+N Q)n

∫

2k+N Q|bj(z) − bj(y)|s′

j dμ(y))

1s′

j , if j ∈ σ′\{j1, j2, . . . , jλ}

thus for j ∈ σ′,

(gj)∗k,s′

j≤ C‖bj‖∗ + C|m2N Q(bj) − bj(z)| + C|m2k+N Q(bj) − bj(z)|.

Hence we get from (3.24) that, for τ > 1,

(

1μ(R)

∫

R

|Iα,m(g1F1, g2F2, . . . , gmFm)(z)|τ dμ(z))

1τ

≤ C

⎛

⎝

m∏

j=1

M( α

m )sj , 9

8fj(x)

⎞

⎠

∞∑

k=1

2−k(n− αm )λ

×⎛

⎝

1μ(R)

∫

R

∣

∣

∣

∣

∣

∣

∏

j∈σ′(gj)∗

k,s′j

∣

∣

∣

∣

∣

∣

τ

dμ(z)

⎞

⎠

1τ

304 X. Tao and T. Zheng NoDEA

≤ C

⎛

⎝

m∏

j=1

M( α

m )sj , 9

8fj(x)

⎞

⎠

∞∑

k=1

2−k(n− αm )λ

∏

j∈σ′‖bj‖∗

× (

1 + KR,2N Q + KR,2k+N Q

)

≤ C

⎛

⎝

m∏

j=1

M( α

m )sj , 9

8fj(x)

⎞

⎠

∏

j∈σ′‖bj‖∗ (3.25)

where we have used the fact that the cubes R and 2NQ are comparable, whichimplies KR,2N Q ≤ C and KR,2k+N Q ≤ C(1 + k). Using the inequality (3.25)above and the identity (3.23), we obtain that, for τ > 1,

A2 ≤ 1μ(R)

∫

R

(|B1(z)| + |B2(z)|) dμ(z)

≤ Cm∑

i=1

∑

σ∈Cmi

⎛

⎝

∏

j∈σ

KQ,R‖bj‖∗

⎞

⎠

×(

1μ(R)

∫

R

∣

∣

∣

[

�bσ′ , Iα,m

]

(�fχRd\2N Q)(z)∣

∣

∣

τ

dμ(z))

1τ

≤ C

m∑

i=1

∑

σ∈Cmi

⎛

⎝

∏

j∈σ

KQ,R‖bj‖∗

⎞

⎠Mτ, 32

([

�bσ′ , Iα,m

]

(�f))

(x)

+C

m∏

j=1

KQ,R‖bj‖∗M( α

m )sj , 9

8fj(x).

The estimates of A3 and A4 is very similar to the one used in the estimateof A2. In fact, repeating the similar procedures used in (3.24) and (3.25) forτ = 1, and noting that KQ,2k+N Q ≤ KQ,R + KR,2k+N Q ≤ C(1 + k) + KQ,R

since 2N−3Q ⊂ 2R ⊂ 2NQ by the definition of N , we can deduce that∑

α1,α2,...,αm∈{0,∞}at least one αi =∞

and one αj =0

∣

∣

∣mQ

[

Iα,m

(

(m˜Q(b1)−b1)fα1

1 , . . . , (m˜Q(bm)−bm)fαm

m

)]∣

∣

∣

+∑

α1,α2,...,αm∈{R,∞}at least one αi =∞

|mR [Iα,m ((mR(b1) − b1)fα11 , . . . ,

× (mR(bm) − bm)fαmm )]|

≤ C

m∏

j=1

KQ,R‖bj‖∗M( α

m )sj , 9

8fj(x). (3.26)

It is left to estimate the term in A3 of the case α1 = α2 = · · · = αm = 0.A small modification is needed to estimate this term. For z ∈ Q and x ∈ Q,one sees

Iα,m

(

(m˜Q(b1) − b1)f0

1 , . . . , (m˜Q(bj) − bj)f0

j , . . . , (m˜Q(bm) − bm)f0

m

)

Vol. 18 (2011) Multilinear commutators with non-doubling measures 305

≤ CN∑

k=1

m∏

j=1

1

l(

2k 43Q

)n− αm

∫

2k 43 Q

|(m˜Q(bj) − bj(yj))fj(yj)|dμ(yj)

≤ CN∑

k=1

m∏

j=1

1

l(

2k 43Q

)n− αm

(

∫

2k 43 Q

|(m˜Q(bj) − bj)|s′

j dμ(yj)

)1

s′j

×(

∫

2k 43 Q

|fj |sj dμ(yj)

)1

sj

≤ C

N∑

k=1

m∏

j=1

μ(

2k 32Q

)1− αmn

l(

2k 43Q

)n− αm

(

1 + KQ, ˜Q + KQ,2k 43 Q

)

‖bj‖∗M( α

m )sj , 9

8fj(x)

≤ C(KQ,RKαQ,R)m

m∏

j=1

‖bj‖∗M( α

m )sj , 9

8fj(x).

This and the inequality (3.26) follows

A3 + A4 ≤ C(KQ,RK(α)Q,R)m

m∏

j=1

‖bj‖∗M( α

m )sj , 9

8fj(x).

Moreover, combing the estimates of A1, A2, A3 and A4, we obtain the desiredinequality (3.11).

Finally, let us show how to acquire the inequality (3.9) from the twoinequalities (3.10) and (3.11). Fix the point x and let Q be any cube such thatx ∈ Q, notice KQ, ˜Q ≤ C and K

(α)

Q, ˜Q≤ C, hence we see from the inequalities

(3.10) and (3.11) that

1μ(

32Q

)

∫

Q

∣

∣

∣

[

�b, Iα,m

]

(�f)(z) − m˜Q

([

�b, Iα,m

]

(�f))∣

∣

∣ dμ(z)

≤ 1μ(

32Q

)

∫

Q

∣

∣

∣

[

�b, Iα,m

]

(�f)(z) − hQ

∣

∣

∣ dμ(z) +1

μ(

32Q

)

∫

Q

∣

∣

∣hQ − h˜Q

∣

∣

∣ dμ(z)

+1

μ(

32Q

)

∫

Q

∣

∣

∣m˜Q

[

�b, Iα,m

]

(�f) − h˜Q

∣

∣

∣ dμ(z)

≤ CR(�b, �f)(x). (3.27)

On the other hand, for all doubling cubes Q ⊂ R with x ∈ Q such thatKQ,R ≤ K

(α)Q,R ≤ P0, where P0 is the constant in Lemma 6 in [3], using (3.11),

we have

|hQ − hR| ≤ CK(α)Q,RP 2m−1

0 R(�b, �f)(x), (3.28)

306 X. Tao and T. Zheng NoDEA

and moreover the inequality (3.28) holds for any doubling cubes Q,R withQ ⊂ R. Therefore,

∣

∣

∣mQ

([

�b, Iα,m

]

(�f))

− mR

([

�b, Iα,m

]

(�f))∣

∣

∣

≤∣

∣

∣mQ

([

�b, Iα,m

]

(�f))

− hQ

∣

∣

∣ +∣

∣

∣hR − mR

([

�b, Iα,m

]

(�f))∣

∣

∣

+ |hQ − hR|≤ CK

(α)Q,RR(�b, �f)(x). (3.29)

According to the estimates (3.27), (3.29) and the definition of the sharp max-imal function, we deduce the inequality (3.9) and so finish the proof of thelemma. �

Corollary 3.5. Using the same notations in Lemma 3.4, and letting σ ⊂{1, 2, . . . ,m}, we have

M �,(α)([

�bσ, Iα,m

]

(�f))

(x) ≤ C

⎡

⎣

∏

j∈σ

‖bj‖∗Mτ, 32(Iα,m(�f))(x)

+∑

σ1∪σ2=σσ1 =∅,σ2 =∅

∏

j∈σ1

‖bj‖∗Mτ, 32

([−→bσ2 , Iα,m

]

(�f))

(x)

+∏

j∈σ

‖bj‖∗m∏

j=1

M( α

m )sj , 9

8fj(x)

⎤

⎦ (3.30)

and particular,

M �,(α)([bi, Iα,m](�f))(x) ≤ C‖bi‖∗

⎡

⎣Mτ, 32(Iα,m(�f))(x) +

m∏

j=1

M( α

m )sj , 9

8fj(x)

⎤

⎦ .

(3.31)

Proof. The proof of (3.30) and (3.31) is very similar to (3.9), so we omit thedetails. �

Now we are ready to give the proof of theorem 1.4. Using Lemma 3.2,Lemma 3.3, Corollary 3.5 and Theorem 1.2, we get that

∥

∥

∥[bi, Iα,m](�f)∥

∥

∥

Mpq (μ)

≤ C∥

∥

∥N([bi, Iα,m](�f))∥

∥

∥

Mpq (μ)

≤ C∥

∥

∥M �,(α)([bi, Iα,m](�f))∥

∥

∥

Mpq (μ)

≤ C‖bi‖∗

∥

∥

∥

∥

∥

Mτ, 32(Iα,m(�f))(x) +

m∏

i=1

M( α

m )si,

98

fi

∥

∥

∥

∥

∥

Mpq (μ)

≤ C‖bi‖∗∥

∥

∥Mτ, 32(Iα,m(�f))

∥

∥

∥

Mpq (μ)

+ C‖bi‖∗

Vol. 18 (2011) Multilinear commutators with non-doubling measures 307

×∥

∥

∥

∥

∥

m∏

i=1

M( α

m )si,

98

fi

∥

∥

∥

∥

∥

Mpq (μ)

≤ C‖bi‖∗∥

∥

∥Iα,m(�f)∥

∥

∥

Mpq (μ)

+ C‖bi‖∗m∏

i=1

‖fi‖Mpiqi (μ)

≤ C‖bi‖∗m∏

j=1

‖fj‖Mpjqj (μ)

.

Similarly, applying the inequality (3.30) in Corollary 3.5, for σ ⊆ {1, 2, . . . ,m},we have

∥

∥

∥[�bσ, Iα,m](�f)∥

∥

∥

Mpq (μ)

≤ C‖N([�bσ, Iα,m](�f))‖Mpq (μ)

≤ C‖M �,(α)[�bσ, Iα,m](�f)‖Mpq (μ)

≤ C

∥

∥

∥

∥

∥

∥

∏

j∈σ

‖bj‖∗Mτ, 32(Iα,m(�f))

+∑

σ1∪σ2=σσ1 =∅,σ2 =∅

∏

j∈σ1

‖bj‖∗Mτ, 32

([−→bσ2 , Iα,m

]

(�f))

+∏

j∈σ

‖bj‖∗m∏

j=1

M( α

m )sj , 9

8fj

∥

∥

∥

∥

∥

∥

Mpq (μ)

≤ C∏

j∈σ

‖bj‖∗∥

∥

∥Mτ, 32(Iα,m(�f))

∥

∥

∥

Mpq (μ)

+C∑

σ1∪σ2=σσ1 =∅,σ2 =∅

∏

j∈σ1

‖bj‖∗∥

∥

∥Mτ, 32

([−→bσ2 , Iα,m

]

(�f))∥

∥

∥

Mpq (μ)

+C∏

j∈σ

‖bj‖∗

∥

∥

∥

∥

∥

∥

m∏

j=1

M( α

m )sj , 9

8fj

∥

∥

∥

∥

∥

∥

Mpq (μ)

≤ C∏

j∈σ

‖bj‖∗m∏

j=1

‖fj‖Mpjqj (μ)

+C∑

σ1∪σ2=σσ1 =∅,σ2 =∅

∏

j∈σ1

‖bj‖∗∥

∥

∥Mτ, 32

([−→bσ2 , Iα,m

]

(�f))∥

∥

∥

Mpq (μ)

≤ C∏

j∈σ

‖bj‖∗m∏

j=1

‖fj‖Mpjqj (μ)

+C∑

σ1∪σ2=σσ1 =∅,σ2 =∅

∏

j∈σ1

‖bj‖∗∥

∥

∥

[−→bσ2 , Iα,m

]

(�f)∥

∥

∥

Mpq (μ)

308 X. Tao and T. Zheng NoDEA

where σ1 and σ2 are two nonempty subsets of σ and σ1 ∩ σ2 = ∅. Hence, wecan make use of induction on σ ⊆ {1, 2, . . . ,m} to get that

∥

∥

∥

[

�bσ, Iα,m

]

(�f)∥

∥

∥

Mpq (μ)

≤ C∏

j∈σ

‖bj‖∗m∏

i=1

‖fi‖Mpiqi (μ).

This completes the proof of Theorem 1.4.

Acknowledgments

The authors thank the anonymous referee for reading the paper carefully andgiving several useful suggestions.

References

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[3] Chen, W., Sawyer, E.: A note on commutators of fractional integral withRBMO(µ) function. Il. J. Math. 46(4), 1287–1298 (2002)

[4] Sawano, Y., Tanaka, H.: Morrey spaces for non-doubling measures. Acta Math.Sin. English Ser. 21, 1535–1544 (2005)

[5] Sawano, Y., Tanaka, H.: Sharp maximal inequality and commutators on Morreyspaces with non-doubling measures. Taiwanese J. Math. 11(4), 1091–1112 (2007)

[6] Hu, G., Meng, Y., Yang, D.: Multilinear commutators of singular integrals withnon-doubling measures. Integr. Equ. Oper. Theory 51, 235–255 (2005)

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X. Tao, T. ZhengDepartment of Mathematics, Zhejiang University of Science and Technology,Hangzhou 310023, People‘s Republic of Chinae-mail: xxtao@hotmail.com

T. Zhenge-mail: taotzheng@126.com

Received: 17 August 2010.

Accepted: 20 December 2010.