Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008, Article ID 373050, 17 pages 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, huoxwu@xmu.edu.cn Received 3 March 2008; Accepted 16 July 2008 Recommended by Nikolaos Papageorgiou Let μ be a nondoubling measure on R d . A class of commutators associated with multilinear fractional integrals and RBMOμ functions are introduced and shown to be bounded on product of Lebesgue spaces with μ. Copyright q 2008 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 see 1–4, and so forth. Meanwhile, as a further development, harmonic analysis on R d 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 see 5–10,etc.. Motivated by 5, 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 in 11 . Before stating our results, we recall some definitions and notations. Let μ be a Radon measure on R d satisfying the following growth condition; there exist constants C>0and n ∈ 0,d, such that μQ ≤ ClQ n , 1.1 for any cube Q ⊂ R d with sides parallel to the coordinate axes, where lQ stands for the side length of Q. For r>0, rQ will denote the cube with the same center as Q and with lrQrlQ. 2 Journal of Inequalities and Applications Let 0 ≤ β<n, given two cubes Q ⊂ R in R d ,weset K β Q,R  1  N Q,R  k1  μ  2 k Q  l  2 k Q  n  1−β/n , 1.2 where N Q,R is the first integer k such that l2 k Q ≥ lR.Ifβ  0, then K 0 Q,R  K Q,R .Thelater quantity was introduced by Tolsa in 11. Given β d depending on d large enough e.g., β d > 2 n , we say that a cube Q ⊂ R d is doubling if μ2Q ≤ β d μQ. Given a cube Q ⊂ R d ,letN be the smallest nonnegative integer such that 2 N Q is doubling. We denote this cube by  Q. Let η>1 be a fixed constant. We say that b ∈ L 1 loc μ is in RBMOμ if there exists a constant C 1 such that for any cube Q 1 μηQ  Q   by − m  Q b   dμy ≤ C 1 ,   m Q b − m R b   ≤ C 1 K Q,R , for any two doubling cubes Q ⊂ R, 1.3 where m Q b  μQ −1  Q bydμy. The minimal constant C 1 is the RBMOμ norm of b,andit will be denoted by b ∗ .In11, Tolsa obtained equivalent norm in the space RBMOμ with different parameters η>1andβ d > 2 n . We consider the following multilinear fractional integral operator I α,m  f 1 , ,f m  x  R d  m f 1  x − y 1  f 2  x − y 2  ···f m  x − y m     y 1 ,y 2 , ,y m    mn−α dμ  y 1  ···dμ  y m  . 1.4 For m  1, we denote I α,1 by I α , which is the Riesz potential operator related to μ. Given m ∈ N, for all 1 ≤ j ≤ m, we denote by C m j the family of all finite subsets σ  {σ1, ,σj} of {1, 2, ,m} of j different elements. For any σ ∈ C m j , we denote σ   {1, 2, ,m}\σ  {σ  j  1, ,σ  m}. Moreover, for b j ∈ RBMOμ, j  1, 2, ,m,let  b b 1 ,b 2 , ,b m  and denote by  b σ b σ1 , ,b σj  and by b σ xb σ1 x ···b σj x. Also, we denote  f f 1 , ,f m ,  f σ f σ1 , ,f σj ,  b σ   f σ  b σ  j1 f σj1 , ,b σ  m f σ  m . We define a kind of commutator of I α,m as follows:   b, I α,m   f  x m  j0  σ∈C m j −1 m−j b σ xI α,m   f σ ,  b σ   f σ   x. 1.5 In particular, for m  2, we define  b 1 ,b 2 ,I α,2  f 1 ,f 2  xb 1 xb 2 xI α,2  f 1 ,f 2  x − b 1 xI α,2  f 1 ,b 2 f 2  x − b 2 xI α,2  b 1 f 1 ,f 2  xI α,2  b 1 f 1 ,b 2 f 2  x. 1.6 Obviously, for m  1, the operator defined in 1.5 is the Coifman-Rochberg-Weiss type commutator of fractional integral, b, I α . Under the assumption that μ is a nondoubling measure, Chen and Sawyer 5 established the L p ,L q -boundedness of b, I α also see 9 for the more general case. In this paper, we will extend the result of 5 as follows. J. Lian and H. Wu 3 Theorem 1.1. Let μ be defined as above and μ  ∞, b j ∈ RBMOR d , j  1, 2, 0 <α<2n.Then b 1 ,b 2 ,I α,2  is a bounded operator from L q 1 × L q 2 to L q with 1/q  1/q 1  1/q 2 − α/2n>0 and 1 <q 1 , q 2 < ∞. Remark 1.2. By Lemma 2.2 in Section 2, Theorem 1.1 for the case μ < ∞ also holds provided I α,2 , b 1 ,b 2 ,I α,2 , b 1 ,I α,2 ,andb 2 ,I α,2  satisfy certain T1 type conditions. For instance, if I α,2 satisfies the T1 condition, that is, I ∗1 α,2  0, then we can easily obtain  I α,2 f 1 ,f 2 xdμx0 see 3 for the notation I ∗1 α,2 . More generally, we have the following theorem. Theorem 1.3. Let m ∈ N, μ be defined as above, and μ  ∞, b j ∈ RBMOR d , j  1, 2, ,m, 0 <α<mn.Then     b, I α,m   f    L q μ ≤ C m  j1   b j   ∗   f j   L q j μ , 1.7 where 1/q  1/q 1  1/q 2  ··· 1/q m − α/mn > 0 and 1 <q j < ∞, j  1, 2, ,m. Clearly, 5, Theorem 1 is the special case of our Theorem 1.3 for m  1. Throughout this paper, we always use the letter C to denote a positive constant that may vary at each occurrence but is independent of the essential variable. 2. Proofs of theorems We only prove Theorem 1.1 since Theorem 1.3 can follow from the same arguments and an analogous version of the following Lemma 2.5, which can be deduced by induction on m. Before proving our results, we need to recall some notation and establish some lemmas which play important roles in the proofs. Let f be a function in L 1 loc R d , we define the noncentered maximal operator M β p,η fxsup Qx  1 μηQ 1−βp/n  Q   fy   p dμy  1/p , 2.1 and the sharp maximal function M #,β fxsup Qx 1 μ  3/2Q   Q   fy − m  Q f   dμy sup R⊃Qx Q,R doubling   m Q f − m R f   K β Q,R , 2.2 where the supremum is taken over all cubes Q with sides parallel to the coordinate axes, m Q f is the mean value of f on the cube Q. When β  0, we denote M 0 p,η f by M p,η f and M #,0 f by M # f. We also consider the noncentered doubling maximal operator N, defined by Nfx sup Qx Q doubling 1 μQ  Q   fy   dμy. 2.3 4 Journal of Inequalities and Applications Lemma 2.1 see 11. Let 1 ≤ p<∞ and 1 <ρ<∞.Thenb ∈ RBMOμ, if and only if for any cube Q ⊂ R d , 1 μρQ  Q   bx − m  Q b   p dμx ≤ Cb p ∗ , 2.4 and for any doubling cubes Q ⊂ R,   m Q b − m R b   ≤ CK Q,R b ∗ . 2.5 Lemma 2.2 see 5. Let f ∈ L 1 loc μ with  fdμ  0 if μ < ∞. For 1 <p<∞,ifinf1, Nf ∈ L p μ, then for 0 ≤ β<nwe have Nf L p μ ≤ C   M #,β f   L p μ . 2.6 Lemma 2.3 see 5. Let p<r<n/αand 1/q  1/r − α/n.Then   M α p,η f   L q μ ≤ Cf L r μ , 2.7 where η>1 and 0 ≤ α<n/p. Lemma 2.4. Suppose μ is a Radon measure satisfying 1.1.Letm ∈ N and 1/s  1/r 1 ···1/r m − α/n > 0 with 0 <α<mn, 1 ≤ r j ≤∞. Then, a if each r j > 1,   I α,m  f 1 , ,f m    L s μ ≤ C m  j1   f j   L r j μ ; 2.8 b if r j  1 for some j,   I α,m  f 1 , ,f m    L s,∞ μ ≤ C m  j1   f j   L r j μ . 2.9 Proof. The proof follows the idea that, for the classical setting, can be found in 4. For the sake of completeness, we will show it again. Since α>0, some r i < ∞.Ifsay,r l1  ···  r m  ∞,1 ≤ l<m, because α/n < 1/r 1  ··· 1/r l ≤ l,sothatmn − α>m − ln, integration in y l1 , ,y m reduces matters to the case when all r i are finite and m  l. Thus, we may assume that all r i < ∞. Now, observe that if 0 <c i , i  1, ,m,and0<α<  m i1 c i , we can find 0 <α i <c i such that α   m i1 α i . Apply this observation to c i  n/r i ,and1/s i  1/r i −α i /n. Since  m i1 1/s i  1/s,0<α i /n ≤ 1, 1 <s i < ∞,and   y 1   n−α 1   y 2   n−α 2 ···   y m   n−α m ≤    y 1 , ,y m    nm−α , 2.10 where α   m i1 α i . It follows that I α,m  f 1 , ,f m  x ≤ m  i1 I α i  f i  x. 2.11 Then, by 5, Lemma 1, page 1289or see 6, page 1269 and H ¨ older’s inequality see 12, page 15 for weak spaces when some r i  1, we can get Lemma 2.4. J. Lian and H. Wu 5 Lemma 2.5. Let b 1 ,b 2 ,I α,2  be as in 1.6, 0 <α<2n, τ>1, b 1 ,b 2 ∈ RBMOμ. Then there exists a constant C>0 such that for all f 1 ∈ L q 1 μ, f 2 ∈ L q 2 μ, and x ∈ R d , M #,α  b 1 ,b 2 ,I α,2  f 1 ,f 2  x ≤ C    b 1   ∗   b 2   ∗ M τ,3/2  I α,2  f 1 ,f 2  x    b 1   ∗ M τ,3/2  b 2 ,I α,2  f 1 ,f 2  x    b 2   ∗ M τ,3/2  b 1 ,I α,2  f 1 ,f 2  x    b 1   ∗   b 2   ∗ M α p 1 ,9/8 f 1 xM α p 2 ,9/8 f 2 x  , 2.12 M #,α  b 1 ,I α,2  f 1 ,f 2  x ≤ C   b 1   ∗  M τ,3/2  I α,2  f 1 ,f 2  x  M α p 1 ,9/8  f 1  xM α p 2 ,9/8  f 2  x  , 2.13 M #,α  b 2 ,I α,2  f 1 ,f 2  x ≤ C   b 2   ∗  M τ,3/2  I α,2  f 1 ,f 2  x  M α p 1 ,9/8  f 1  xM α p 2 ,9/8  f 2  x  , 2.14 where  b 1 ,I α,2  f 1 ,f 2  xb 1 xI α,2  f 1 ,f 2  x − I α,2  b 1 f 1 ,f 2  x,  b 2 ,I α,2  f 1 ,f 2  xb 2 xI α,2  f 1 ,f 2  x − I α,2  f 1 ,b 2 f 2  x. 2.15 Proof. By the definition, to obtain 2.12,itsuffices to prove that for any x ∈ R d and a cube Q  x, 1 μ  3/2Q   Q    b 1 ,b 2 ,I α,2  f 1 ,f 2  z − h Q   dμz≤C    b 1   ∗   b 2   ∗ M τ,3/2  I α,2  f 1 ,f 2  x    b 1   ∗ M τ,3/2  b 2 ,I α,2   f 1 ,f 2  x    b 2   ∗ M τ,3/2  b 1 ,I α,2   f 1 ,f 2  x    b 1   ∗   b 2   ∗ M α p 1 ,9/8 f 1 xM α p 2 ,9/8 f 2 x  , 2.16 and for any cubes Q ⊂ R, where Q is an arbitrary cube and R is doubling,   h Q − h R   ≤ CK 2 Q,R K α Q,R    b 1   ∗   b 2   ∗ M τ,3/2  I α,2  f 1 ,f 2  x    b 1   ∗ M τ,3/2  b 2 ,I α,2  f 1 ,f 2  x    b 2   ∗ M τ,3/2  b 1 ,I α,2  f 1 ,f 2  x    b 1   ∗   b 2   ∗ M α p 1 ,9/8 f 1 xM α p 2 ,9/8 f 2 x  , 2.17 where h Q  m Q  I α,2  m  Q  b 1  − b 1  f 1 χ R d \4/3Q ,  m  Q  b 2  − b 2  f 2 χ R d \4/3Q  , h R  m R  I α,2  m  R  b 1  − b 1  f 1 χ R d \4/3R ,  m  R  b 2  − b 2  f 2 χ R d \4/3R  . 2.18 6 Journal of Inequalities and Applications First of all, it is easy to see that    b 1 ,b 2 ,I α,2  f 1 ,f 2  z − h Q   ≤    b 1 z − m  Q  b 1  b 2 z − m  Q  b 2  I α,2  f 1 ,f 2  z       b 1 z − m  Q  b 1  I α,2  f 1 ,  b 2 z − b 2  f 2  z       b 2 z − m  Q  b 2  I α,2  b 1 z − b 2  f 1 ,f 2  z      I α,2  b 1 − m  Q b 1  f 1 ,  b 1 − m  Q b 1  f 2  z − h Q   : IzIIzIIIzIVz. 2.19 Consequently, 1 μ  3/2Q   Q    b 1 ,b 2 ,I α,2  f 1 ,f 2  z − h Q   dμz ≤ CI  II  III  IV, 2.20 where I  μ3/2Q −1  Q Izdμz, 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 μ  3/2Q   Q Izdμz ≤ C  1 μ  3/2Q   Q   b 1 z − m  Q  b 1    τ 1 dμz  1/τ 1 ×  1 μ  3/2Q   Q   b 2 z − m  Q  b 2    τ 2 dμz  1/τ 2 ×  1 μ  3/2Q   Q   I α,2  f 1 ,f 2    τ dμz  1/τ ≤ C   b 1   ∗   b 2   ∗ M τ,3/2  I α,2  f 1 ,f 2  x, 2.21 where τ 1 > 1, τ 2 > 1and1/τ  1/τ 1  1/τ 2  1. For II, we have II  1 μ  3/2Q   Q IIzdμz ≤ C  1 μ  3/2Q   Q   b 1 z − m  Q  b 1    s dμz  1/s ×  1 μ  3/2Q   Q    b 2 ,I α,2  f 1 ,f 2  z   τ dμz  1/τ ≤ C   b 1   ∗ M τ,3/2  b 2 ,I α,2  f 1 ,f 2  x, 2.22 where s>1and1/s  1/τ  1. J. Lian and H. Wu 7 Similarly, we have III ≤ C   b 2   ∗ M τ,3/2  b 2 ,I α,2  f 1 ,f 2  x. 2.23 It remains to estimate IV. For convenience, we set f 0 j  f j χ 4/3Q , f j  f 0 j  f ∞ j , j  1, 2. Then,   IVz   ≤   I α,2  b 1 − m  Q  b 1  f 0 1 ,  b 2 − m  Q  b 2  f 0 2  z      I α,2  b 1 − m  Q  b 1  f 0 1 ,  b 2 − m  Q  b 2  f ∞ 2  z      I α,2  b 1 − m  Q  b 1  f ∞ 1 ,  b 2 − m  Q  b 2  f 0 2  z      I α,2  b 1 − m  Q  b 1  f ∞ 1 ,  b 2 − m  Q  b 2  f ∞ 2  z − h Q    IV 1 zIV 2 zIV 3 zIV 4 z, 2.24 and so we have 1 μ  3/2Q   Q   IVz   dμz ≤ 4  j1 1 μ  3/2Q   Q IV j zdμz : 4  j1 IV j . 2.25 To estimate IV 1 ,sets 1  √ p 1 , s 2  √ p 2 ,and1/v  1/s 1  1/s 2 −α/n. It follows from H ¨ older’s inequality and Lemma 2.4 that IV 1 ≤ μQ 1−1/v μ  3/2Q    I α,2  b 1 − m  Q  b 1  f 0 1 ,  b 2 − m  Q  b 2  f 0 2    L v μ ≤ Cμ  3/2Q  −1/v    b 1 − m  Q  b 1  f 0 1   L s 1 μ    b 2 − m  Q  b 2  f 0 2   L s 2 μ ≤ C μ  3/2Q  1/v   4/3Q   f 1 y 1    p 1 dμy 1   1/p 1 ×   4/3Q   b 1 y 1  − m  Q  b 1    p 1 / √ p 1 −1 dμy 1    √ p 1 −1/p 1 ×   4/3Q   f 2 y 2    p 2 dμy 2   1/p 2   4/3Q   b 2 y 1  − m  Q  b i    p 2 / √ p 2 −1 dμy i    √ p 2 −1/p 2 ≤ C 2  i1  1 μ  3/2Q  1−αp i /2n  4/3Q   f i y i    p i dμy i   1/p i ×  1 μ  3/2Q   4/3Q   b i y i  − m  Q  b i    p i / √ p i −1 dμy i    √ p i −1/p i ≤ C   b 1   ∗   b 2   ∗ M α p 1 ,9/8 f 1 xM α p 2 ,9/8 f 2 x. 2.26 8 Journal of Inequalities and Applications For term IV 2 ,byLemma 2.1, we have IV 2  1 μ  3/2Q   Q IV 2 zdμz ≤ C 1 μ  3/2Q   Q  R d \4/3Q  4/3Q ×    b 1  y 1  − m  Q  b 1  f 0 1  y 1       b 2  y 2  − m  Q  b 2  f ∞ 2  y 2       z − y 1 ,z− y 2    2n−α dμ  y 1  dμ  y 2  dμz ≤ C μ  3/2Q   Q  4/3Q    b 1 y − m  Q  b 1  f 0 1  y 1    dμ  y 1  ×  R d \4/3Q    b 2  y 2  − m  Q  b 2  f ∞ 2  y 2      z − y 2   2n−α dμ  y 2  dμz ≤ C  1 μ  3/2Q  1−αp 1 /2n  4/3Q   f 1  y 1    p 1 dμ  y 1   1/p 1 ×  1 μ  3/2Q   4/3Q   b 1 − m  Q  b 1    p  1 dμ  y 1   1/p  1 × μ  3 2 Q  −α/2n μQ ∞  k1  2 k 4/3Q\2 k−1 4/3Q    b 2  y 2  − m  Q  b 2  f 2  y 2    2 k2n−α lQ 2n−α dμ  y 2  ≤ C   b 1   ∗ M α p 1 ,9/8 f 1 x ∞  k1 2 −kn−α/2 l  2 k 3 2 Q  −nα/2 ×  2 k 4/3Q    b 2  y 2  − m  Q  b 2  f 2  y 2    dμ  y 2  ≤ C   b 1   ∗ M α p 1 ,9/8 f 1 x ∞  k1 2 −kn−α/2 l  2 k 3 2 Q  −nα/2 ×   2 k 4/3Q    b 2  y 2  − m  2 k 4/3Q  b 2  f 2  y 2    dμ  y 2     m  2 k 4/3Q  b 2  − m  Q  b 2     2 k 3/2Q   f 2  y 2    dμ  y 2   ≤ C   b 1   ∗ M α p 1 ,9/8 f 1 x ×  ∞  k1 2 −kn−α/2  1 l  2 k 3/2Q  n  2 k 4/3Q    b 2  y 2  − m  2 k 4/3Q  b 2    p  2 dμ  y 2   1/p  2 ×  1 l  2 k 3/2Q  n−αp 2 /2  2 k 4/3Q   f 2  y 2    p 2 dμ  y 2   1/p 2  ∞  k1 k2 −kn−α/2   b 2   ∗ 1 l  2 k 3/2Q  n−α/2  2 k 4/3Q   f 2  y 2    dμ  y 2   ≤ C   b 1   ∗   b 2   ∗ M α p 1 ,9/8 f 1 xM α p 2 ,9/8 f 2 x, 2.27 J. Lian and H. Wu 9 where the last inequality follows from the following two facts: 1 l  2 k 3/2Q  n−α/2  2 k 4/3Q   f 2  y 2    dμ  y 2  ≤ μ  2 k 4/3Q  1−1/p 2 l  2 k 3/2Q  n−α/2   2 k 4/3Q   f 2  y 2    p 2 dμ  y 2   1/p 2 ≤ C μ  2 k1 4/3Q  1−1/p 2 1/p 2 −α/2n l  2 k1 4/3Q  n−α/2  1 μ  2 k1 4/3Q  1−αp 2 /2n  2 k1 4/3Q   f 2  y 2    p 2 dμ  y 2   1/p 2 ≤ CM α p 2 ,9/8 f 2 x, 2.28 and see11   m  2 k 4/3Q  b j  − m  Q  b j    ≤ C   b j   ∗ K  Q,  2 k 4/3Q ≤ C   b j   ∗ K Q,2 k 4/3Q ≤ Ck   b j   ∗ ,j 1, 2. 2.29 Similarly, IV 3 ≤ C   b 1   ∗   b 2   ∗ M α p 1 ,9/8 f 1 xM α p 2 ,9/8 f 2 x. 2.30 For term IV 4 , we have   I α,2  b 1 − m  Q  b 1  f ∞ 1 ,  b 2 − m  Q  b 2  f ∞ 2  z − I α,2  b 1 − m  Q  b 1  f ∞ 1 ,  b 2 − m  Q  b 2  f ∞ 2  y   ≤  R d \4/3Q  R d \4/3Q     1    z − y 1 ,z− y 2    2n−α − 1    y − y 1 ,y− y 2    2n−α     ×     2  i1  b i  y i  − m  Q  b i  f ∞ i  y i      dμ  y 1  dμ  y 2  ≤  R d \4/3Q  R d \4/3Q |z − y|    y − y 1 ,y− y 2    2n−α1 ×     2  i1  b i  y i  − m  Q  b i  f ∞ i  y i      dμ  y 1  dμ  y 2  ≤ C 2  i1  R d \4/3Q |z − y| 1/2   y − y i   n−α/21/2    b i  y i  − m  Q  b i  f ∞ i  y i    dμ  y i  ≤ C 2  i1 ∞  k1  2 k 4/3Q\2 k−1 4/3Q 2 −k/2 1 l  2 k Q  n−α/2    b i  y i  − m  Q  b i      f ∞ i  y i    dμ  y i  ≤ C 2  i1 ∞  k1 2 −k/2  1 l  2 k 3/2Q  n  2 k 4/3Q    b i  y i  − m  Q  b i    p  i dμ  y i   1/p  i ×  1 l  2 k 3/2Q  n−αp i /2  2 k 4/3Q   f i  y i    p i dμ  y i   1/p i 10 Journal of Inequalities and Applications ≤ C 2  i1 ∞  k1 2 −k/2 M α p i ,9/8 f i x ×  1 l  2 k 3/2Q  n  2 k 4/3Q    b i  y i  −m  2 k 4/3Q  b i  m  2 k 4/3Q  b i  −m  Q  b i    p  i dμ  y i   1/p  i ≤ C 2  i1 ∞  k1 2 −k/2 k   b i   ∗ M α p i ,9/8 f i x ≤ C   b 1   ∗   b 2   ∗ M α p 1 ,9/8 f 1 xM α p 2 ,9/8 f 2 x. 2.31 Taking the mean over y ∈ Q,weobtain   I α,2  b 1 −m  Q  b1  f ∞ 1 ,  b 1 −m  Q  b 1  f ∞ 2  z−h Q   ≤ C   b 1   ∗   b 2   ∗ M α p 1 ,9/8 f 1 xM α p 2 ,9/8 f 2 x. 2.32 Thus, IV 4  1 μ  3/2Q   Q IV 4 zdμz ≤ C   b 1   ∗   b 2   ∗ M α p 1 ,9/8 f 1 xM α p 2 ,9/8 f 2 x. 2.33 Combing 2.20–2.33,weobtain2.16. Now we turn to estimate 2.17. For any cubes, Q ⊂ R with x ∈ Q, where Q is arbitrary and R is doubling. We denote N Q,R  1simplybyN, write   h Q − h R      m Q  I α,2  b 1 − m  Q b 1  f ∞ 1 ,  b 2 − m  Q b 2  f ∞ 2  − m R  I α,2  b 1 − m R b 1  f ∞ 1 ,  b 2 − m R b 2  f ∞ 2    ≤   m R  I α,2  b 1 − m  Q b 1  f 1 χ R d \2 N Q ,  b 2 − m  Q b 2  f 2 χ R d \2 N Q  − m Q  I α,2  b 1 − m  Q b 1  f 1 χ R d \2 N Q ,  b 2 − m  Q b 2  f 2 χ R d \2 N Q       m R  I α,2  b 1 − m R b 1  f 1 χ R d \2 N Q ,  b 2 − m R b 2  f 2 χ R d \2 N Q  − m R  I α,2  b 1 − m  Q b 1  f 1 χ R d \2 N Q ,  b 2 − m  Q b 2  f 2 χ R d \2 N Q       m Q  I α,2  b 1 − m  Q b 1  f 1 χ 2 N Q\4/3Q ,  b 2 − m  Q b 2  f 2 χ R d \4/3Q       m Q  I α,2  b 1 − m  Q b 1  f 1 χ R d \2 N Q ,  b 2 − m  Q b 2  f 2 χ 2 N Q\4/3Q       m R  I α,2  b 1 − m R b 1  f 1 χ R d \4/3R ,  b 2 − m R b 2  f 2 χ 2 N Q\4/3R       m R  I α,2  b 1 − m R b 1  f 1 χ 2 N Q\4/3R ,  b 2 − m R b 2  f 2 χ R d \2 N Q     6  i1 A i . 2.34 By the similar arguments used in proving 2.33,weobtainthat A 1 ≤ C  K Q,R  2   b 1   ∗   b 2   ∗ M α p 1 ,9/8 f 1 xM α p 2 ,9/8 f 2 x. 2.35 [...]... 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` tiques, vol 48, no 2, pp 335–367, 2004 a 10 G Hu, Y Meng, and D Yang, Multilinear commutators of singular integrals. .. 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Multilinear Calderon-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 RBMO μ functions,” Illinois Journal of Mathematics, vol 46, no 4, pp 1287–1298, 2002 6 J Garc a- Cuerva... singular integrals with non doubling measures,” Integral Equations and Operator Theory, vol 51, no 2, pp 235–255, 2005 11 X Tolsa, “BMO, H 1 , and Calderon-Zygmund operators for non doubling measures,” Mathematische ´ Annalen, vol 319, no 1, pp 89–149, 2001 12 L Grafakos, Classical and Modern Fourier Analysis, Pearson Education/Prentice Hall, Upper Saddle River, NJ, USA, 2004 ... definition of the sharp maximal function, we complete the proof of 2.12 Similarly, we can deduce 2.13 and 2.14 The details are omitted Now we are in the position to prove Theorem 1.1 Proof of Theorem 1.1 By the Lebesgue differentiation theorem, it is easy to see that for any f ∈ L1 Rd , loc ≤ Nf x , f x 2.59 for μ − a. e x ∈ Rd , see 11 for details By Lemmas 2.2–2.5, we have b1 , b2 , Iα,2 f1 , f2 Lq... Lq Lq J Lian and H Wu 17 ≤C b1 ∗ b2 ∗ f1 Lq1 f2 Lq2 b1 ∗ M#, β b2 , Iα,2 f1 , f2 Lp1 b2 ≤ C b1 ∗ ∗ M#, β b1 , Iα,2 f1 , f2 Lp2 b2 ∗ f1 Lq1 f2 Lq2 2.60 This proves Theorem 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 China G10571122 References 1 C P´ rez and R H Torres,... 9/8 f2 x α On the other hand, for all doubling cubes Q ⊂ R with x ∈ Q, such that KQ,R ≤ Pα , where Pα is the constant in 5, Lemma 6 , by 2.17 , we have 2 hQ − hR ≤ CKQ,R Pα b1 ∗ b2 b1 ∗ Mτ, 3/2 b2 , Iα,2 f1 , f2 x b2 Mτ, 3/2 ∗ b1 , Iα,2 f1 , f2 x b1 ∗ ∗ Mτ, 3/2 Iα,2 f1 , f2 b2 ∗ x Mp1 , 9/8 f1 x Mp2 , 9/8 f2 x 2.56 16 Journal of Inequalities and Applications Hence, by 5, Lemma 6 , we get α hQ − hR ≤... , 9/8 f1 x Mp2 , 9/8 f2 x By the same arguments, we can get α A2 2 ≤ CKQ,R KQ,R b1 Mτ, 3/2 Iα,2 f1 , f2 Mτ, 3/2 ∗ x b2 , Iα,2 f1 , f2 x 2.50 Mp1 , 9/8 f1 x Mp2 , 9/8 f2 x Consequently, α 2 A2 ≤ CKQ,R KQ,R b1 Mτ, 3/2 Iα,2 f1 , f2 b2 ∗ x 2.51 Mp1 , 9/8 f1 x Mp2 , 9/8 f2 x Using the similar arguments to those used in proving B5 z , we can conclude that A3 A4 A5 α 2 A6 ≤ CKQ,R KQ,R b1 ∗ b2 ∗ α α Mp1 ,... f1 χ2N Q , f2 χRd \ 4/3 R z Iα,2 2.39 b1 − mR b1 f1 χ2N Q\ 4/3 R , f2 χ2N Q\ 4/3 R z 7 Bj z j 1 For B1 z , we write Iα,2 b1 − mR b1 f1 , f2 z ≤ Iα,2 b1 − b1 z f1 , f2 z Iα,2 b1 z − mR b1 f1 , f2 z 2.40 12 Journal of Inequalities and Applications By Holder’s inequality and the fact that R is doubling, we have ¨ 1 μ R R 1 μ R R Iα,2 b1 − mR b1 f1 , f2 z dμ z ≤ C b1 ∗ Mτ, 3/2 Iα,2 f1 , f2 x , 2.41 Iα,2 . Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008, Article ID 373050, 17 pages doi:10.1155/2008/373050 Research Article A Class of Commutators for Multilinear. huoxwu@xmu.edu.cn Received 3 March 2008; Accepted 16 July 2008 Recommended by Nikolaos Papageorgiou Let μ be a nondoubling measure on R d . A class of commutators associated with multilinear fractional integrals and RBMOμ. P ´ erez and R. H. Torres, “Sharp maximal function estimates for multilinear singular integrals, ” in Harmonic analysis at Mount Holyoke, vol. 320 of Contemporary Mathematics, pp. 323–331, American Mathematical