A Class of Commutators for Multilinear Fractional Integrals in Nonhomogeneous Spaces

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Volume 2008, Article ID 373050,17pages doi:10.1155/2008/373050

Research Article

A Class of Commutators for Multilinear Fractional

Integrals in Nonhomogeneous Spaces

Jiali Lian and Huoxiong Wu

School of Mathematical Sciences, Xiamen University, Xiamen Fujian, 361005, China

Correspondence should be addressed to Huoxiong Wu,[email protected]

Received 3 March 2008; Accepted 16 July 2008

Recommended by Nikolaos Papageorgiou

Letμ be a nondoubling measure onRd_{. A class of commutators associated with multilinear} fractional integrals and RBMOμfunctions are introduced and shown to be bounded on product of Lebesgue spaces withμ.

Copyrightq2008 J. Lian and H. Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

In recent years, the study of multilinear operators and their commutator has been attracting many researchers. Many results which parallel to the linear theory of classical integral operators are obtained. For details, one can see1–4, and so forth. Meanwhile, as a further development, harmonic analysis on Rd _{with nondoubling measures has been developed} rapidly. Many results of singular integrals and the related operators on Euclidean spaces with Lebesgue measure have been generalized to the Lebesgue spaces with nondoubling measures

see5–10, etc.. Motivated by5,8, we will consider the commutators generated by a class of multilinear fractional integrals and RBMO functions with nondoubling measure, which were introduced by Tolsa in11.

Before stating our results, we recall some definitions and notations. Letμbe a Radon measure onRd _{satisfying the following growth condition; there exist constants} _{C >} _{0 and}

n∈0, d, such that

μQ≤ClQn

, 1.1

for any cubeQ ⊂ Rd _{with sides parallel to the coordinate axes, where} _lQ _{stands for the} side length ofQ. Forr > 0,rQ will denote the cube with the same center as Q and with

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Let 0≤β < n, given two cubesQ⊂RinRd_{, we set}

K_Q,Rβ 1 NQ,R

k1

μ2kQ l2kQn

1−β/n

, 1.2

whereNQ,Ris the first integerksuch thatl2kQ≥lR. Ifβ0, thenK0_Q,RKQ,R. The later quantity was introduced by Tolsa in11.

Givenβd depending ondlarge enoughe.g.,βd>2n, we say that a cubeQ⊂ Rdis doubling ifμ2Q≤βdμQ.

Given a cube Q ⊂ Rd_{, let} _N _{be the smallest nonnegative integer such that 2}N_Q _is doubling. We denote this cube byQ.

Letη > 1 be a fixed constant. We say thatb ∈L1

locμis in RBMOμif there exists a constantC1such that for any cubeQ

1

μηQ

Q

_by₋_m

Qb dμy≤C1,

_m_Q_b₋_m_R_b _≤_C₁_K_Q,R_, _{for any two doubling cubes}_Q_⊂_R, 1.3

wheremQbμQ−1

Qbydμy. The minimal constantC1is the RBMOμnorm ofb, and it will be denoted byb∗. In11, Tolsa obtained equivalent norm in the space RBMOμwith

diﬀerent parametersη >1 andβd>2n.

We consider the following multilinear fractional integral operator

Iα,m

f1, . . . , fm

x

Rdm

f1

x−y1

f2

x−y2

· · ·fm

x−ym

_y₁_{, y}₂_{, . . . , y}_m mn−α dμ

y1

· · ·dμym

. 1.4

Form1, we denoteIα,1byIα, which is the Riesz potential operator related toμ. Given m ∈ N, for all 1 ≤ j ≤ m, we denote by Cm

j the family of all finite subsets

σ {σ1, . . . , σj}of{1,2, . . . , m}ofj diﬀerent elements. For anyσ ∈ C_jm, we denoteσ {1,2, . . . , m} \σ {σj 1, . . . , σm}. Moreover, for bj ∈ RBMOμ, j 1,2, . . . , m, let

b b1, b2, . . . , bmand denote bybσ bσ1, . . . , bσjand by bσx bσ1x· · ·bσjx. Also, we denotef f1, . . . , fm,fσ fσ1, . . . , fσj,bσf_σ b_σ_j₁f_σ_j₁, . . . , b_σ_mf_σ_m. We define a kind of commutator ofIα,mas follows:

b, Iα,m

fx

m

j0

σ∈Cm j

−1m−jbσxIα,m

fσ, bσfσ

x. 1.5

In particular, form2, we define

b1, b2, Iα,2

f1, f2

x b1xb2xIα,2

f1, f2

x−b1xIα,2

f1, b2f2

x −b2xIα,2

b1f1, f2

x Iα,2

b1f1, b2f2

x. 1.6

Obviously, form 1, the operator defined in 1.5is the Coifman-Rochberg-Weiss type commutator of fractional integral,b, Iα. Under the assumption thatμis a nondoubling measure, Chen and Sawyer5established theLp_{, L}q_{-boundedness of}_{b, I}

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Theorem 1.1. Letμbe defined as above andμ∞,bj ∈RBMORd,j1,2,0< α <2n. Then

b1, b2, Iα,2is a bounded operator fromLq1×Lq2 toLq with1/q 1/q11/q2−α/2n > 0and 1< q1,q2 <∞.

Remark 1.2. ByLemma 2.2inSection 2,Theorem 1.1for the caseμ<∞also holds provided

Iα,2,b1, b2, Iα,2,b1, Iα,2, andb2, Iα,2satisfy certainT1type conditions. For instance, ifIα,2 satisfies theT1condition, that is,I∗1

α,2 0, then we can easily obtain

Iα,2f1, f2xdμx 0 see3for the notationI∗1

α,2.

More generally, we have the following theorem.

Theorem 1.3. Letm∈N,μbe defined as above, andμ ∞,bj ∈ RBMORd,j 1,2, . . . , m, 0< α < mn. Then

_{b, I}_α,m

f_Lq_μ≤C

m

j1

_b_j_∗_f_j

Lqjμ, 1.7

where1/q1/q11/q2· · ·1/qm−α/mn >0and1< qj <∞,j 1,2, . . . , m.

Clearly,5, Theorem 1is the special case of ourTheorem 1.3form 1. Throughout this paper, we always use the letterCto denote a positive constant that may vary at each occurrence but is independent of the essential variable.

2. Proofs of theorems

We only proveTheorem 1.1sinceTheorem 1.3can follow from the same arguments and an analogous version of the followingLemma 2.5, which can be deduced by induction onm. Before proving our results, we need to recall some notation and establish some lemmas which play important roles in the proofs.

Letfbe a function inL1_locRd_{, we define the noncentered maximal operator}

M_p,β_ηfx sup Qx

1

μηQ1−βp/n

Q

_fy p_dμy1/p

, 2.1

and the sharp maximal function

M#,βfx sup Qx

1

μ3/2Q

Q

_fy₋_m_Q_f _dμy sup R⊃Qx Q,Rdoubling

_m_Q_f₋_m_R_f K_Q,Rβ ,

2.2

where the supremum is taken over all cubes Q with sides parallel to the coordinate axes,

mQfis the mean value offon the cubeQ. Whenβ0, we denoteM_p,0_ηfbyMp,ηfand M#,0_f_by_M#_f_.

We also consider the noncentered doubling maximal operatorN, defined by

Nfx sup Qx Qdoubling

1

μQ

Q

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Lemma 2.1see11. Let1≤p < ∞and1< ρ <∞. Thenb∈RBMOμ, if and only if for any cubeQ⊂Rd_,

1

μρQ

Q

_bx₋_m_Q_b p_dμx_≤_Cbp

∗, 2.4

and for any doubling cubesQ⊂R,

_m_Q_b₋_m_R_b _≤_CK_Q,R_b_∗_. 2.5 Lemma 2.2see5. Letf ∈L1

locμwith

fdμ0ifμ<∞.For1< p <∞, ifinf1,Nf∈ Lp_μ_{, then for}₀_≤_{β < n}_{we have}

NfLp_μ≤CM#,βf

Lp_μ. 2.6

Lemma 2.3see5. Letp < r < n/αand1/q1/r−α/n. Then _Mα

p,ηfLq_μ≤CfLr_μ, 2.7

whereη >1and0≤α < n/p.

Lemma 2.4. Supposeμis a Radon measure satisfying1.1. Letm∈Nand1/s1/r1· · ·1/rm−

α/n >0with0< α < mn,1≤rj ≤ ∞. Then,

aif eachrj>1,

_I_α,m

f1, . . . , fm_Ls_μ≤C

m

j1 _f_j

Lrjμ; 2.8

bifrj1for somej,

_I_α,m

f1, . . . , fm_Ls,∞_μ≤C m

j1 _f_j

Lrjμ. 2.9

Proof. The proof follows the idea that, for the classical setting, can be found in4. For the sake of completeness, we will show it again.

Sinceα > 0, someri < ∞. If say, rl1 · · · rm ∞, 1 ≤ l < m, because α/n < 1/r1· · ·1/rl ≤l, so thatmn−α >m−ln, integration inyl1, . . . , ymreduces matters to the case when allriare finiteandml. Thus, we may assume that allri<∞. Now, observe that if 0< ci,i 1, . . . , m, and 0 < α < mi1ci, we can find 0 < αi < cisuch thatαmi1αi. Apply this observation tocin/ri, and 1/si 1/ri−αi/n. Sincemi11/si1/s, 0< αi/n≤1, 1< si<∞, and

_y₁ n−α1

y2 n−α2· · · ym n−αm ≤ y1, . . . , ym nm−α, 2.10 whereαmi1αi. It follows that

Iα,m

f1, . . . , fm

x≤m

i1 Iαi

fi

x. 2.11

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Lemma 2.5. Letb1, b2, Iα,2be as in1.6,0< α <2n,τ >1,b1, b2∈RBMOμ. Then there exists a constantC >0such that for allf1∈Lq1μ,f2∈Lq2μ, andx∈Rd,

M#,αb1, b2, Iα,2

f1, f2

x≤Cb1_∗b2_∗Mτ,3/2

Iα,2

f1, f2

x b1_∗Mτ,3/2

b2, Iα,2

f1, f2

x b2_∗Mτ,3/2b1, Iα,2

f1, f2

x

b1_∗b2_∗M_pα₁_,₉_/₈f1xM_pα₂_,₉_/₈f2x

,

2.12

M#,α_b

1, Iα,2

f1, f2

x≤Cb1_∗

Mτ,3/2Iα,2

f1, f2

x

M_pα₁_,₉_/₈f1

xMα

p2,9/8

f2

x,

2.13

M#,αb2, Iα,2

f1, f2

x≤Cb2_∗

Mτ,3/2Iα,2

f1, f2

x

M_pα₁_,₉_/₈f1

xMα

p2,9/8

f2

x,

2.14

where

b1, Iα,2

f1, f2

x b1xIα,2

f1, f2

x−Iα,2

b1f1, f2

x,

b2, Iα,2

f1, f2

x b2xIα,2

f1, f2

x−Iα,2

f1, b2f2

x. 2.15

Proof. By the definition, to obtain2.12, it suﬃces to prove that for anyx∈ Rd _{and a cube}

Qx, 1

μ3/2Q

Q

_b₁_{, b}₂_{, I}_α,₂ f1, f2

z−hQ dμz≤Cb1_∗b2_∗Mτ,3/2Iα,2

f1, f2

x

b1_∗Mτ,3/2b2, Iα,2

f1, f2

x b2_∗Mτ,3/2b1, Iα,2

f1, f2

x

b1_∗b2_∗M_pα₁_,₉_/₈f1xM_pα₂_,₉_/₈f2x

, 2.16

and for any cubesQ⊂R, whereQis an arbitrary cube andRis doubling,

_h_Q₋_h_R _≤_CK2

Q,RK

α

Q,Rb1_∗b2_∗Mτ,3/2

Iα,2

f1, f2

x

b1_∗Mτ,3/2b2, Iα,2

f1, f2

x b2_∗Mτ,3/2b1, Iα,2

f1, f2

x

b1_∗b2_∗M_pα₁_,₉_/₈f1xM_pα₂_,₉_/₈f2x

,

2.17

where

hQ mQ

Iα,2

mQ

b1

−b1

f1χRd_\₄_/₃_Q,

mQ

b2

−b2

f2χRd_\₄_/₃_Q

, hRmR

Iα,2

mR

b1

−b1

f1χRd_\₄_/₃_R,

mR

b2

−b2

f2χRd_\₄_/₃_R

.

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First of all, it is easy to see that

_b₁_{, b}₂_{, I}_α,₂ f1, f2

z−hQ ≤ b1z−mQ

b1

b2z−mQ

b2

Iα,2

f1, f2

z

b1z−mQ

b1

Iα,2

f1,

b2z−b2

f2

z

b2z−mQ

b2

Iα,2

b1z−b2

f1, f2

z Iα,2

b1−mQb1

f1,

b1−mQb1

f2

z−hQ

:Iz IIz IIIz IVz.

2.19

Consequently,

1

μ3/2Q

Q

_b₁_{, b}₂_{, I}_α,₂ f1, f2

z−hQ dμz≤CIIIIIIIV, 2.20

where Iμ3/2Q−1_QIzdμz, and II, III, IV are defined in the same way.

In what follows, we estimate I–IV, respectively. For I, by H ¨older’s inequality and

Lemma 2.1, we have

I 1

μ3/2Q

Q

Izdμz

≤C

1

μ3/2Q

Q

_b₁_z₋_m

Q

b1 τ1dμz 1/τ1

×

1

μ3/2Q

Q

_b₂_z₋_m_Q

b2 τ2dμz 1/τ2

×

1

μ3/2Q

Q

_I_α,₂

f1, f2 τdμz 1/τ

≤Cb1_∗b2_∗Mτ,3/2

Iα,2

f1, f2

x,

2.21

whereτ1>1,τ2>1 and 1/τ1/τ11/τ21. For II, we have

II 1

μ3/2Q

Q

IIzdμz

≤C

1

μ3/2Q

Q

_b₁_z₋_m_Q

b1 sdμz 1/s

× 1 μ3/2Q

Q

_b₂_{, I}_α,₂ f1, f2

z τ_dμz1/τ

≤Cb1_∗Mτ,3/2b2, Iα,2

f1, f2

x,

2.22

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Similarly, we have

III≤Cb2_∗Mτ,3/2b2, Iα,2

f1, f2

x. 2.23

It remains to estimate IV. For convenience, we setf_j0 fjχ4/3Q,fj f_j0f_j∞,j 1,2. Then,

IVz ≤ Iα,2

b1−mQ

b1

f₁0,b2−mQ

b2

f₂0z Iα,2

b1−mQ

b1

f₁0,b2−mQ

b2

f2∞

z

Iα,2

b1−mQ

b1

f₁∞,b2−mQ

b2

f₂0z Iα,2

b1−mQ

b1

f1∞,

b2−mQ

b2

f2∞

z−hQ

IV1z IV2z IV3z IV4z,

2.24

and so we have

1

μ3/2Q

Q

_IV_z _dμz_≤4

j1 1

μ3/2Q

Q

IVjzdμz:

4

j1

IVj. 2.25

To estimate IV1, sets1√p1,s2 √p2, and 1/v1/s11/s2−α/n. It follows from H ¨older’s inequality andLemma 2.4that

IV1≤ μQ

1−1/v

μ3/2QIα,2

b1−mQ

b1

f₁0,b2−mQ

b2

f₂0_Lv_μ

≤Cμ3/2Q−1/vb1−mQ

b1

f0

1_Ls1μb2−mQ

b2

f0

2_Ls2μ

≤ C

μ3/2Q1/v

4/3Q

_f₁_y₁ p1_dμy

1 1/p1

×

4/3Q

_b₁_y₁₋_m_Q b1 p1/

√_p

1−1_dμy

1

√p1−1/p1

×

4/3Q

_f₂_y₂ p2_dμy

2

1/p2

4/3Q

_b₂_y₁₋_m_Q bi p2/

√_p

2−1_dμy i

√p2−1/p2

≤C 2

i1

1

μ3/2Q1−αpi/2n

4/3Q

_f_i_y_i pi_dμy

i

1/pi

×

1

μ3/2Q

4/3Q

_b_i_y_i₋_m_Q bi pi/

√_p

i−1_dμy

i

√_p

i−1/pi

≤Cb1_∗b2_∗M_pα₁_,₉_/₈f1xM_pα₂_,₉_/₈f2x.

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For term IV2, byLemma 2.1, we have

IV2 1

μ3/2Q

Q

IV2zdμz

≤C 1 μ3/2Q

Q

Rd_\₄_/₃_Q

4/3Q

× b1

y1

−mQ b1 f0 1

y1 b2

y2

−mQ

b2

f₂∞y2 _z₋_y₁_{, z}₋_y₂ 2n−α dμ

y1

dμy2

dμz

≤ C

μ3/2Q

Q

4/3Q

_b₁_y₋_m_Q b1

f₁0y1 dμ

y1

×

Rd_\₄_/₃_Q

_b₂ y2

−mQ

b2

f₂∞y2 _z₋_y₂ 2n−α dμ

y2 dμz ≤C 1

μ3/2Q1−αp1/2n

4/3Q

_f₁

y1 p1dμ

y1 1/p1

×

1

μ3/2Q

4/3Q

_b₁₋_m_Q b1 p

1_dμ_y

1 1/p₁

×μ

3 2Q

−α/2n

μQ∞

k1

2k₄_/₃_Q_\₂k−1₄_/₃_Q

_b₂ y2

−mQ

b2 f2 y2 2k2n−α_lQ2n−α dμ

y2

≤Cb1_∗M_pα₁_,₉_/₈f1x

∞

k1

2−kn−α/2_l

2k3 2Q

₋nα/2

×

2k₄_/₃_Q

_b₂ y2

−mQ

b2

f2

y2 dμ

y2

≤Cb1_∗M_pα₁_,₉_/₈f1x

∞

k1

2−kn−α/2l

2k3 2Q

−nα/2

×

2k₄_/₃_Q

_b₂ y2

−m₂k₄_/₃_Q

b2

f2

y2 dμ

y2

m₂k₄_/₃_Q

b2

−mQ

b2

2k₃_/₂_Q

_f₂ y2 dμ

y2

≤Cb1_∗M_pα₁_,₉_/₈f1x

× _∞

k1

2−kn−α/2

1

l2k3/2Qn

2k₄_/₃_Q

_b₂ y2

−m₂k₄_/₃_Q

b2 p

2_dμ_y

2 1/p2

×

1

l2k3/2Qn−αp2/2

2k₄_/₃_Q

_f₂

y2 p2dμ

y2 1/p2

∞

k1

k2−kn−α/2b2_∗

1

l2k3/2Qn−α/2

2k₄_/₃_Q

_f₂ y2 dμ

y2

≤Cb1_∗b2_∗M_pα₁_,₉_/₈f1xM_pα₂_,₉_/₈f2x,

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where the last inequality follows from the following two facts:

1

l2k3/2Qn−α/2

2k₄_/₃_Q

_f₂ y2 dμ

y2

≤ μ

2k4/3Q1−1/p2

l2k3/2Qn−α/2

2k₄_/₃_Q

_f₂

y2 p2dμ

y2 1/p2

≤Cμ

2k14/3Q1−1/p21/p2−α/2n

l2k14/3Qn−α/2

1

μ2k14/3Q1−αp2/2n

2k1₄_/₃_Q

_f₂

y2 p2dμ

y2 1/p2

≤CM_pα₂_,₉_/₈f2x,

2.28 andsee11

_m

2k₄_/₃_Q

bj

−mQ

bj ≤Cbj_∗K_Q,₂k₄_/₃_Q≤Cbj∗KQ,2k₄_/₃_Q≤Ckb_j_∗, j1,2.

2.29 Similarly,

IV3 ≤Cb1_∗b2_∗M_pα₁_,₉_/₈f1xM_pα₂_,₉_/₈f2x. 2.30 For term IV4, we have

_I_α,₂ b1−mQ

b1

f₁∞,b2−mQ

b2

f2∞

z−Iα,2

b1−mQ

b1

f₁∞,b2−mQ

b2

f2∞

y

≤

Rd_\₄_/₃_Q

1

z−y1, z−y2 2n−α

− 1

y−y1, y−y2 2n−α

× 2

i1

bi

yi

−mQ

bi

f_i∞yi dμ

y1

dμy2

≤

Rd_\₄_/₃_Q

|z−y|

_y₋_y₁_{, y}₋_y₂ 2n−α1

× 2

i1

bi

yi

−mQ

bi

f_i∞yi dμ

y1

dμy2

≤C 2

i1

Rd_\₄_/₃_Q

|z−y|1/2 _y₋_y_i n−α/21/2 bi

yi

−mQ

bi

f_i∞yi dμ

yi

≤C 2

i1

∞

k1

2k₄_/₃_Q_\₂k−1₄_/₃_Q2

−k/2 1

l2kQn−α/2 _b_i

yi

−mQ

bi fi∞

yi dμ

yi

≤C 2

i1

∞

k1 2−k/2

1

l2k3/2Qn

2k₄_/₃_Q

_b_i yi

−mQ

bi p

i_dμ_y

i

1/pi

×

1

l2k3/2Qn−αpi/2

2k₄_/₃_Q

_f_i

yi pidμ

yi

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≤C 2

i1

∞

k1

2−k/2M_pα

i,9/8fix

×

1

l2k3/2Qn

2k₄_/₃_Q

_b_i yi

−m₂k₄_/₃_Q

bi

m₂k₄_/₃_Q

bi

−mQ

bi p

i_dμ_y

i

1/pi

≤C 2

i1

∞

k1

2−k/2kbi_∗M_pα_i_,₉_/₈fix

≤Cb1_∗b2_∗M_pα₁_,₉_/₈f1xM_pα₂_,₉_/₈f2x.

2.31

Taking the mean overy∈Q, we obtain

_I_α,₂ b1−mQ

b1f1∞,

b1−mQ

b1

f2∞

z−hQ ≤Cb1_∗b2_∗M_pα₁_,₉_/₈f1xM_pα₂_,₉_/₈f2x. 2.32

Thus,

IV4 1

μ3/2Q

Q

IV4zdμz≤Cb1_∗b2_∗M_pα₁_,₉_/₈f1xM_pα₂_,₉_/₈f2x. 2.33

Combing2.20–2.33, we obtain2.16.

Now we turn to estimate2.17. For any cubes,Q⊂Rwithx∈Q, whereQis arbitrary andRis doubling. We denoteNQ,R1 simply byN, write

_h_Q₋_h_R _m_Q Iα,2

b1−mQb1

f₁∞,b2−mQb2

f₂∞ −mR

Iα,2

b1−mRb1

f₁∞,b2−mRb2

f2∞ ≤ mR

Iα,2

b1−mQb1

f1χRd_\₂N_Q,

b2−mQb2

f2χRd_\₂N_Q

−mQ

Iα,2

b1−mQb1

f1χRd_\₂N_Q,

b2−mQb2

f2χRd_\₂N_Q

mR

Iα,2

b1−mRb1

f1χRd_\₂N_Q,

b2−mRb2

f2χRd_\₂N_Q

−mR

Iα,2

b1−mQb1

f1χRd_\₂N_Q,

b2−mQb2

f2χRd_\₂N_Q

mQ

Iα,2

b1−mQb1

f1χ2N_Q_\₄_/₃_Q,

b2−mQb2

f2χRd_\₄_/₃_Q

mQ

Iα,2

b1−mQb1

f1χRd_\₂N_Q,

b2−mQb2

f2χ2N_Q_\₄_/₃_Q

mR

Iα,2

b1−mRb1

f1χRd_\₄_/₃_R,

b2−mRb2

f2χ2N_Q_\₄_/₃_R

mR

Iα,2

b1−mRb1

f1χ2N_Q_\₄_/₃_R,b₂−m_Rb₂f₂χ_Rd_\₂N_Q

6

i1 Ai.

2.34

By the similar arguments used in proving2.33, we obtain that

A1 ≤C

KQ,R

2

(11)

To estimateA2, we write

Iα,2

b1−mR

b1

f1χRd_\₂N_Q,

b2−mR

b2

f2χRd_\₂N_Q

z −Iα,2

b1−mQ

b1

f1χRd_\₂N_Q,

b2−mQ

b2

f2χRd_\₂N_Q

z mR b2 −mQ

b2

Iα,2

b1−mR

b1

f1χRd_\₂N_Q, f₂χ_Rd_\₂N_Q

z mR b1 −mQ

b1

Iα,2

f1χRd_\₂N_Q,

b2−mR

b2

f2χRd_\₂N_Q

z mR b1 −mQ

b1 mR b2 −mQ

b2

Iα,2

z.

2.36

Then,

A2≤ mR

b2

−mQ

b2

_μR1

R

Iα,2

b1−mR

b1

zdμz mR b1 −mQ

b1

_μR1

R

Iα,2

f1χRd_\₂N_Q,b₂−m_Rb₂f₂χ_Rd_\₂N_Qzdμz

mR b1 −mQ

b1 mR

b2

−mQ

b2

_μR1

R

Iα,2

zdμz

A21A22A23.

2.37

It is obvious that

A23≤CK_Q,R2 b1_∗b2_∗Mτ,3/2Iα,2

f1, f2

x. 2.38

In order to estimate termA21, we write

Iα,2

b1−mRb1

z Iα,2

b1−mRb1

f1, f2

z−Iα,2

b1−mRb1

f1χ2N_Qχ₄_/₃_R, f₂χ₄_/₃_R

z −Iα,2

b1−mRb1

f1χ4/3R, f2χ2N_Qχ₄_/₃_R

z Iα,2

b1−mRb1

f1χ2N_Qχ₄_/₃_R, f₂χ₂N_Qχ₄_/₃_Rz

−Iα,2

b1−mRb1

f1χRd_\₄_/₃_R, f₂χ₂N_Qz

−Iα,2

b1−mRb1

f1χ2N_Q, f₂χ_Rd_\₄_/₃_R

z Iα,2

b1−mRb1

f1χ2N_Q_\₄_/₃_R, f₂χ₂N_Q_\₄_/₃_R

z

7

j1 Bjz.

2.39

ForB1z, we write _I_α,₂

b1−mRb1

f1, f2

z ≤ Iα,2

b1−b1z

f1, f2

z Iα,2

b1z−mRb1

f1, f2

(12)

By H ¨older’s inequality and the fact thatRis doubling, we have

1

μR

R

Iα,2

b1−mRb1

f1, f2

zdμz≤Cb1_∗Mτ,3/2

Iα,2

f1, f2

x,

1

μR

R

Iα,2

b1−b1z

f1, f2

zdμz≤CMτ,3/2b1, Iα,2

f1, f2

x,

2.41

which imply

_m_R_B₁ _≤_C_b₁

∗Mτ,3/2

Iα,2

f1, f2

x Mτ,3/2

b1, Iα,2

f1, f2

x. 2.42

ForB2z, sets1√p1,s2p2, and 1/v1/s11/s2−α/n. Using H ¨older’s inequality and

Lemma 2.4, we have

1

μR

R

B2zdμz≤C

μR1−1/v

μR Iα,2

b1−mR

b1

f1χ2N_Qχ₄_/₃_R, f₂χ₄_/₃_R

Lv_μ

≤Cμ

3 2R

−1/v

_b₁₋_m_R b1

f1χ2N_Qχ₄_/₃_R

Ls1_μf2χ4/3RLs2μ

≤Cμ

3 2R

₋1/v

4/3R

_f₂_y p2_dμy

1/p2

4/3R

_f₁_x p1_dμy

1/p1

×

4/3R

_b₁_y₋_m_R b1 p1/

√_p

1−1_dμy

√p1−1/p1

≤C

1

μ3/2R1−αp1/2n

4/3R

_f₁_y p1_dμy

1/p1

×

1

μ3/2R

4/3R

_b₁_y₋_m_R b1 p1/

√_p

1−1_dμy

√_p 1−1/p1

×

1

μ3/2R1−αp2/2n

4/3Q

_f₂_y p2_dμy

1/p2

≤Cb1_∗M_pα₁_,₉_/₈f1xM_pα₂_,₉_/₈f2x,

2.43

which implies

_m_R_B₂ _≤_C_b₁_∗_Mα

p1,9/8f1xM

α

p2,9/8f2x. 2.44 Similarly,

_m_R_B₃ _≤_C_b₁

∗Mpα1,9/8f1xM

α

p2,9/8f2x,

_m_R_B₄ _≤_C_b₁

∗Mpα1,9/8f1xM

α

p2,9/8f2x.

(13)

ForB5z, sincez∈R, we have

_B₅_z _≤ 2N_Q

Rd_\₄_/₃_Q

_b₁₋_m_R b1

f1 f2

y2 _z₋_y₁_{, z}₋_y₂ 2n−α dμ

y1

dμy2

≤

2N_Q

_f₂ y2 dμ

y2

Rd_\₄_/₃_Q

_b₁ y1

−mR

b1

f1 _z₋_y₁ 2n−α dμ

y1

≤ C l2Rn−α/2

2N_Q

_f₂ y2 dμ

y2

×∞

k1

2−kn

l2k4/3n−α/2

2k₄_/₃_R_\₂k−1₄_/₃_R

_b₁ y1

−mR

b1

f1 dμ

y1

≤C 1 l2Rn−α/2

2N_Q

_f₂ y2 dμ

y2

×∞

k1

2−kn

l2k4/3Rn−α/2

×

2k₄_/₃_R

_b₁ y1

−m2k₄_/₃_R

b1

f1 dμ

y1

2k₄_/₃_R

_m₂k₄_/₃_R

b1

−mR

b1

y1 f1

y1 dμ

y1

≤C 1 l2Rn−α/2

2N_Q

_f₂ y2 dμ

y2

×∞

k1 2−kn

1

l2k4/3Rn

2k₄_/₃_R

_b₁ y1

−m2k₄_/₃_R

b1 p

1_dμ_y

1 1/p₁

× 1

l2k4/3Rn−αp1/2

2k₄_/₃_R

_f₁ p1

dμy1 1/p1

Cb1_∗ 1

l2k4/3Rn−α/2

2k₄_/₃_R

_f₁ _dμ y1

≤C 1 l2Rn−α/2

2N_Q

_f₂ y2 dμ

y2

∞

k1

2−knb1_∗

M_pα₁_,₉_/₈f1x Mp1,9/8f1x

≤C

N

k1 1

l2Rn−α/2

2k_Q_\₂k−1_Q

_f₂ y2 dμ

y2b1_∗M_pα₁_,₉_/₈f1x

1

l2Rn−α/2

Q

_f₂ y2 dμ

y2b1_∗M_pα₁_,₉_/₈f1x

≤C

N

k0 1

l2Rn−α/2

2k_Q

_f₂ y2 dμ

(14)

≤C

N

k0

μ2k1Q1−α/2n l2k1Qn−α/2

l2k1Qn−α/2 l2Rn−α/2

1

μ2k1Q1−α/2n

×

2k_Q

_f₂ y2 dμ

y2b1_∗M_pα₁_,₉_/₈f1x

≤C

N

k0

μ2k1Q1−α/2n l2k1Qn−α/2

1

μ9/82k_Q1−α/2n

2k_Q

_f₂ y2 dμ

y2b1_∗M_pα₁_,₉_/₈f1x

≤CK_Q,Rαb1_∗M_pα₁_,₉_/₈f1xM_pα₂_,₉_/₈f2x.

2.46

Taking the mean onzoverR, we obtain

_m_R_B₅ _≤_CKα

Q,Rb1∗Mpα1,9/8f1xM

α

p2,9/8f2x. 2.47 Similarly, we have

_m_R_B₆ _≤_CKα

α

p2,9/8f2x,

_m_R_B₇ _≤_CKα

α

p2,9/8f2x.

2.48

Summing up the estimates2.39–2.48, we obtain

A21≤CKQ,RK_Q,Rαb1_∗b2_∗Mτ,3/2Iα,2

f1, f2

x

b2_∗Mτ,3/2b1, Iα,2

f1, f2

x b1_∗b2_∗Mp1,9/8f1xMp2,9/8f2x

.

2.49

By the same arguments, we can get

A22≤CKQ,RKQ,Rαb1_∗b2_∗Mτ,3/2

Iα,2

f1, f2

x

b1_∗Mτ,3/2b2, Iα,2

f1, f2

x b1_∗b2_∗Mp1,9/8f1xMp2,9/8f2x

.

2.50

Consequently,

A2≤CKQ,R2 K α

Q,Rb1∗b2_∗Mτ,3/2Iα,2

f1, f2

x

b1_∗Mτ,3/2b2, Iα,2

f1, f2

x b2_∗Mτ,3/2b1, Iα,2

f1, f2

x b1_∗b2_∗Mp1,9/8f1xMp2,9/8f2x

.

2.51

Using the similar arguments to those used in provingB5z, we can conclude that

A3A4A5A6 ≤CK2Q,RK

α

(15)

Therefore, we obtain

_h_Q₋_h_R _≤_CK2

Q,RK

α

Q,Rb1_∗b2_∗Mτ,3/2

Iα,2

f1, f2

x

b1_∗Mτ,3/2b2, Iα,2

f1, f2

x b2_∗Mτ,3/2b1, Iα,2

f1, f2

x b1_∗b2_∗Mp1,9/8f1xMp2,9/8f2x

,

2.53

which implies2.17.

Finally, we show how to derive 2.12 from 2.16 and 2.17. From 2.16, if Q is doubling andx∈Q, we have

_m_Q

b1, b2, Iα,2

f1, f2

−hQ ≤ 1

μQ

Q

_b₁_{, b}₂_{, I}_α,₂ f1, f2

z−hQ dμz

≤Cb1_∗b2_∗Mτ,3/2Iα,2

f1, f2

x b1_∗Mτ,3/2b2, Iα,2

f1, f2

x b2_∗Mτ,3/2b1, Iα,2

f1, f2

x b1_∗b2_∗Mp1,9/8f1xMp2,9/8f2x

.

2.54

Also, for any cubeQwithx∈ Q,K_Q,Q ≤ CandK α

Q,Q ≤ C, by2.16,2.17, and2.54, we have

1

μ3/2Q

Q

_b₁_{, b}₂_{, I}_α,₂ f1, f2

z−mQ

b1, b2, Iα,2

f1, f2 dμz

≤ 1

μ3/2Q b1, b2, Iα,2

f1, f2

z−hQ dμz

1

μ3/2Q

Q

_m_Q

b1, b2, Iα,2

f1, f2

−hQ dμz

1

μ3/2Q

Q

_h_Q₋_h_Q _dμz

≤Cb1_∗b2_∗Mτ,3/2Iα,2

f1, f2

x b1_∗Mτ,3/2b2, Iα,2

f1, f2

x b2_∗Mτ,3/2b1, Iα,2

f1, f2

x b1_∗b2_∗Mp1,9/8f1xMp2,9/8f2x

.

2.55

On the other hand, for all doubling cubesQ⊂Rwithx∈Q, such thatK_Q,Rα ≤Pα, wherePα is the constant in5, Lemma 6, by2.17, we have

_h_Q₋_h_R _≤_CK2

Q,RPαb1_∗b2_∗Mτ,3/2Iα,2

f1, f2

x b1_∗Mτ,3/2b2, Iα,2

f1, f2

x b2_∗Mτ,3/2b1, Iα,2

f1, f2

x b1_∗b2_∗Mp1,9/8f1xMp2,9/8f2x

.

(16)

Hence, by5, Lemma 6, we get

_h_Q₋_h_R _≤_CKα

Q,Rb1∗b2_∗Mτ,3/2Iα,2

f1, f2

x

b1_∗Mτ,3/2b2, Iα,2

f1, f2

x b2_∗Mτ,3/2

b1, Iα,2

f1, f2

x b1_∗b2_∗Mp1,9/8f1xMp2,9/8f2x

,

2.57

for all doubling cubesQ⊂Rwithx∈Q. Invoking2.55yields that

_m_Q

b1, b2, Iα,2

f1, f2

−mR

b1, b2, Iα,2

f1, f2 ≤ mQ

b1, b2, Iα,2

f1, f2

−hQ hR−mR

b1, b2, Iα,2

f1, f2 hQ−hR

≤CK_Q,Rαb1_∗b2_∗Mτ,3/2Iα,2

f1, f2

x

b1_∗Mτ,3/2b2, Iα,2

f1, f2

x b2_∗Mτ,3/2b1, Iα,2

f1, f2

x b1_∗b2_∗Mp1,9/8f1xMp2,9/8f2x

.

2.58

From this estimate and the definition of the sharp maximal function, we complete the proof of2.12. Similarly, we can deduce2.13and2.14. The details are omitted.

Now we are in the position to proveTheorem 1.1.

Proof ofTheorem 1.1. By the Lebesgue diﬀerentiation theorem, it is easy to see that for any

f∈L1 locRd,

_fx _{≤ Nfx,} 2.59

forμ− a.e.x∈Rd_{, see}₁₁_{for details. By Lemmas}_2.2_–_2.5_{, we have}

_b₁_{, b}₂_{, I}_α,₂

f1, f2_Lq≤N

b1, b2, Iα,2

f1, f2_Lq

≤M#,βb1, b2, Iα,2

f1, f2_Lq

≤Cb1_∗b2_∗Mτ,3/2Iα,2

f1, f2_Lq

b1_∗Mτ,3/2b2, Iα,2

f1, f2_Lq

b2_∗Mτ,3/2b1, Iα,2

f1, f2_Lq

b1_∗b2_∗M_pα₁_,₉_/₈f1xM_pα₂_,₉_/₈f2_Lq

≤Cb1_∗b2_∗f1_Lq₁f2_Lq₂

b1_∗b2, Iα,2

f1, f2_Lq

b2_∗b1, Iα,2

f1, f2_Lq

(17)

≤Cb1_∗b2_∗f1_Lq1f2Lq2

b1_∗M#,β

b2, Iα,2

f1, f2_Lp₁

b2_∗M#,β

b1, Iα,2

f1, f2_Lp2

≤Cb1_∗b2_∗f1_Lq1f2Lq2.

2.60

This provesTheorem 1.1.

Acknowledgments

The authors would like to express their deep thanks to the referees for their valuable remarks and suggestions. Huoxiong Wu was partially supported by the NSF of ChinaG10571122.

References

1 C. P´erez and R. H. Torres, “Sharp maximal function estimates for multilinear singular integrals,” inHarmonic analysis at Mount Holyoke, vol. 320 ofContemporary Mathematics, pp. 323–331, American Mathematical Society, Providence, RI, USA, 2003.

2 C. P´erez and R. Trujillo-Gonz´alez, “Sharp weighted estimates for multilinear commutators,”Journal of the London Mathematical Society, vol. 65, no. 3, pp. 672–692, 2002.

3 L. Grafakos and R. H. Torres, “Multilinear Calder ´on-Zygmund theory,”Advances in Mathematics, vol. 165, no. 1, pp. 124–164, 2002.

4 C. E. Kenig and E. M. Stein, “Multilinear estimates and fractional integration,”Mathematical Research Letters, vol. 6, no. 1, pp. 1–15, 1999.

5 W. Chen and E. Sawyer, “A note on commutators of fractional integrals with RBMOμfunctions,” Illinois Journal of Mathematics, vol. 46, no. 4, pp. 1287–1298, 2002.

6 J. Garc´ıa-Cuerva and J. Mar´ıa-Martell, “Two-weight norm inequalities for maximal operators and fractional integrals on non-homogeneous spaces,”Indiana University Mathematics Journal, vol. 50, no. 3, pp. 1241–1280, 2001.

7 J. Xu, “Boundedness of multilinear singular integrals for non-doubling measures,” Journal of Mathematical Analysis and Applications, vol. 327, no. 1, pp. 471–480, 2007.

8 J. Xu, “Boundedness in Lebesgue spaces for commutators of multilinear singular integrals and RBMO functions with non-doubling measures,”Science in China. Series A, vol. 50, no. 3, pp. 361–376, 2007.

9 G. Hu, Y. Meng, and D. Yang, “Multilinear commutators for fractional integrals in non-homogeneous spaces,”Publicacions Matem`atiques, vol. 48, no. 2, pp. 335–367, 2004.

10 G. Hu, Y. Meng, and D. Yang, “Multilinear commutators of singular integrals with non doubling measures,”Integral Equations and Operator Theory, vol. 51, no. 2, pp. 235–255, 2005.

11 X. Tolsa, “BMO,H1_{, and Calder ´on-Zygmund operators for non doubling measures,”}_{Mathematische}

Annalen, vol. 319, no. 1, pp. 89–149, 2001.