Compactness for the commutators of multilinear singular integral operators with non-smooth kernels

Transcription

1 Appl. Math. J. Chinese Univ. 209, 34(: Compactness for the commutators of multilinear singular integral operators with non-smooth kernels BU Rui CHEN Jie-cheng,2 Abstract. In this paper, the behavior for commutators of a class of bilinear singular integral operators associated with non-smooth kernels on the product of weighted Lebesgue spaces is considered. By some new maximal functions to control the commutators of bilinear singular integral operators and CMO(R n functions, compactness for the commutators is proved. Introduction In recent decades, the study of multilinear analysis becomes an active topic in harmonic analysis. The first important work, among several pioneer papers, is the famous work by Coifman and Meyer in [8,9], where they established a bilinear multiplier theorem on the Lebesgue spaces. Note that a multilinear multiplier actually is a convolution operator. Naturally one will study the non-convolution operator T (f,..., f m (x = K(x, y,..., y m f (y f m (y m dy dy m, (R n m ( where K(x, y,..., y m is a locally integral function defined away from the diagonal x = y = = y m in (R n m+, x / m supp f j and f,..., f m are bounded functions with compact supports. Precisely, T : S(R n S(R n S (R n is an m-linear operator associated with the kernel K(x, y,..., y m. If there exist positive constants A and γ (0, ] such that K satisfies the size condition A K(x, y,..., y m ( x y + + x y m mn (2 Received: Revised: MR Subject Classification: 42B20, 42B25. Keywords: singular integral operator, maximal function, weighted norm inequality, commutator, compact operator. Digital Object Identifier(DOI: Supported by the Natural Science Foundation of Shandong Province(Nos. ZR208PA004 and ZR206AB07, the National Natural Science Foundation of China(Nos and *Corresponding author.

2 56 Appl. Math. J. Chinese Univ. Vol. 34, No. for all (x, y,..., y m (R n m+ with x y j for some j {, 2,..., m}; and the smoothness condition K(x, y,..., y m K(x A x x γ, y,..., y m ( m i= x y i, mn+γ whenever x x 2 max j m x y j and also for each j, K(x, y,..., y j,..., y m K(x, y,..., y j, A y j y j..., y m γ ( m i= x y, (3 mn+γ i whenever y j y j 2 max j m x y j, then we say that K is a Calderón-Zygmund kernel and denote it by K m CZK(A, γ. Also, T is called the multilinear Calderón-Zygmund operator associated with the kernel K. In [6], Grafakos and Torres established the multilinear T theorem, so that they obtained the strong type boundedness on product of L p (R n spaces and endpoint weak type estimates of operators T associated with kernels K m CZK(A, γ. Furthermore, the A p (R n weights (see Definition.2 on the operator T and on the corresponding maximal operator were considered in [5]. After then, the study of multilinear Calderón-Zygmund operator is fruitful. The readers can refer to [4-6,2,23,25-28,32] and the references therein. However, there are some multilinear singular integral operators, including the Calderón commutator, whose kernels do not satisfy (3. Here, the Calderón commutator is defined by m C m+ (f, a,..., a m (x = (A j(x A j (y R (x y m+ f(ydy, where A j = a j for all j {, 2,, m}. This operator first appeared in the study of Cauchy integral along Lipschitz curves and, in fact, led to the first proof of the L 2 -boundedness of the latter. Moreover, the operator C m+ can be viewed as an (m + -linear operator. Define { : x > 0, e(x = 0 : x < 0. Since A j = a j, the (m + -linear operator C m+ (f, a,..., a m can be written as C m+ (f, a,..., a m (x = K(x, y,..., y m+ a (y a m (y m f(y m+ dy dy m+, (R m+ where the kernel K is m K(x, y,..., y m+ = ( e(ym+ xm (x y m+ m+ χ (min(x,ym+,max(x,y m+(y j. (4 In [], the authors pointed out that the kernel K in (4 does not satisfy (3. In order to discuss these operators whose kernels don t satisfy (3, Duong et al. [0,] introduced a class of multilinear singular integral operators whose kernels satisfy the following assumptions (H,(H2 and (H3. Let {A t } t>0 be a class of integral operators, which play the role of an approximation to the identity as in [2]. We always assume that the operators A t are associated with kernels a t (x, y in the sense that A t f(x = a t (x, yf(ydy R n

3 BU Rui, CHEN Jie-cheng. Compactness for the commutators of singular integral operators 57 for all f L p (R n, p, and the kernels a t (x, y satisfy a t (x, y h t (x, y = t n/s x y h(, (5 t /s where s is a fixed positive constant and h is a bounded, positive, decreasing function satisfying for some ι > 0. lim r rn+ι h(r s = 0 (6 Assumption (H. Assume that for each j {, 2,, m}, there exist operators {A (j t } t>0 with kernels a (j t (x, y that satisfy conditions (5 and (6 with constants s and ι, and there exist kernels K (j t (x, y,, y m such that < T (f,, A (j t f j,, f m, g >= R n K (j t (R n m (x, y,, y m f (y f m (y m g(xd ydx for all f,, f m, g S(R n with m supp f j supp g =, and there exist a function Φ C(R with suppφ [, ] and a constant ɛ > 0 such that for all x, y,, y m R n and t > 0, we have K(x, y,, y m K (j C t (x, y,, y m ( x y + + x y m mn for some C > 0, whenever 2t /s x y j. + Ct ɛ/s ( x y + + x y m mn+ɛ m,k j Φ( y j y k t /s Kernels K satisfying condition (2 and assumption (H with parameters m, C, s, ι, ɛ are called generalized Calderón-Zygmund kernels, and their collection is denoted by m GCZK 0 (C, s, ι, ɛ. It was proved in [] that the assumption (H is weaker than condition (3. But it is sufficient for the weak type endpoint estimate. Recall that the j-th transpose T j of T is defined via < T j (f,, f m, h >=< T (f,, f j, h, f j+,, f m, f j > for all f,, f m, h S(R n. It is easy to check that the kernel K j of T j is related to the kernel K of T via the identity K j (x, y,, y j, y j, y j+,, y m = K(y j, y,, y j, x, y j+,, y m. Notice that if a multilinear operator T maps a product of Banach spaces X X m to another Banach space X, then T j maps the product of Banach spaces X X j X X j+ X m to Banach space X j. Moreover, the norms of T and T j are equal. For notational convenience, we may occasionally denote T by T 0 and K by K 0. Assumption (H2. Assume that for every i {,, m}, there exist operators {A (i t } t>0 with kernels a (i t (x, y that satisfy conditions (5 and (6 with constants s and ι, and there also exist kernels K j,(i t (x, y,, y m so that < T j (f,, A (i t f i,, f m, g >= R n (R n m K j,(i t (x, y,, y m m f k (y k g(xd ydx for each j {0,,, m}, and all f,, f m S(R n with m supp f k supp g =. Also assume that there exist a function Φ C(R with suppφ [, ] and a constant ɛ > 0 so that

4 58 Appl. Math. J. Chinese Univ. Vol. 34, No. for all x, y,, y m R n, t > 0, each j {0,,, m}, and every i {,, m}, we have K j (x, y,, y m K j,(i C m t (x, y,, y m ( x y + + x y m mn Φ( y i y k t /s whenever 2t /s x y i. + Ct ɛ/s ( x y + + x y m mn+ɛ,,k i Kernels K satisfying condition (2 and assumption (H2 with parameters m, C, s, ι, ɛ are also called generalized Calderón-Zygmund kernels, and their collection is denoted by m GCZK(C, s, ι, ɛ. We say that T is of class m GCZO(C, s, ι, ɛ if the kernel of it is in m GCZK(C, s, ι, ɛ. In [], the authors obtained the strong type boundedness on product of L p (R n spaces and endpoint weak type estimates of operators T m GCZO(C, s, ι, ɛ. Furthermore, they pointed out that the m-th order Calderón commutator C m+ is an (m + - linear operator associated with a kernel K in (m + GCZK(C,,,. In this way they first proved that for p,..., p m+ [, ] and p (0, with p = m+ and if min j m+ p j >, then C m+ (f, a,..., a m L p, (R f L p m+ (R C m+ (f, a,..., a m Lp (R f L p m+ (R m m p j, a j L p j (R, a j L p j (R. Assumption (H3. Assume that there exist operators {B t } t>0 with kernels b t (x, y that satisfy conditions (5 and (6 with constants s and ι, and there also exist kernels K (0 t (x, y,, y m such that K (0 t (x, y,, y m = K(z, y,, y m b t (x, zdz R n for all x, y,, y m R n. Also assume that there exist a function Φ C(R with suppφ [, ] and a constant ɛ > 0 so that for all x, y,, y m R n and t > 0, we have K(x, y,, y m K (0 C t (x, y,, y m ( x y + + x y m mn + Ct ɛ/s ( x y + + x y m mn+ɛ m Φ( x y k t /s for some C > 0, whenever 2t /s max j m x y j. Moreover, assume that for all x, y,, y m R n, K (0 C t (x, y,, y m ( x y + + x y m mn, whenever 2t /s min j m x y j, and for all x, x, y,, y m R n, t (x, y,, y m K (0 t (x Ct ɛ/s, y,, y m ( x y + + x y m mn+ɛ, K (0 whenever 2 x x t /s and 2t /s min j m x y j. In [0], under assumption (H2 and (H3, some weighted estimates of operator T are obtained, and the authors also proved that the m-th order Calderón commutator C m+ satisfies

5 BU Rui, CHEN Jie-cheng. Compactness for the commutators of singular integral operators 59 assumption (H3. Furthermore, the weighted estimates, including the multiple weights, of the maximal Calderón commutator were considered in [0] and [4]. Moreover, there are a large amount of work related to singular integral operators with non-smooth kernels. The readers may refer [2], [8] and [9], among many interesting works. In this article, we are interested in the compactness for the commutators of multilinear singular integral operators with non-smooth kernels and CMO(R n functions, where CMO(R n denotes the closure of Cc (R n in the BMO(R n topology. For the sake of convenience, we will write out the case of compactness in a bilinear setting. In particular, We will study the compactness for the commutator of T, where we assume that T is a bilinear singular integral operator associated with kernel K in the sense of ( and satisfying (2, and (i T is bounded from L (R n L (R n L /2, (R n ; (7 (ii for x, x, y, y 2 R n with 8 x x < min j 2 x y j, K(x, y, y 2 K(x Dτ γ, y, y 2, (8 ( x y + x y 2 2n+γ where D is a constant and τ is a number such that 2 x x < τ and 4τ < min j 2 x y j. It was pointed out in [20] that the condition (8 is weaker than, and indeed a consequence of, assumption (H3. For b BMO(R n, we consider commutators T b (f, f 2 = [b, T ] (f, f 2 = bt (f, f 2 T (bf, f 2, T 2 b (f, f 2 = [b, T ] 2 (f, f 2 = bt (f, f 2 T (f, bf 2. For b = (b, b 2 BMO(R n BMO(R n, we consider the iterated commutator T b (f, f 2 = [b 2, [b, T ] ] 2 (f, f 2, and, in the sense of (, [b, T ] (f, f 2 (x = K(x, y, y 2 (b(x b(y f (y f 2 (y 2 dy dy 2, R n R n [b, T ] 2 (f, f 2 (x = K(x, y, y 2 (b(x b(y 2 f (y f 2 (y 2 dy dy 2, R n R n T b (f, f 2 (x = K(x, y, y 2 (b (x b (y (b 2 (x b 2 (y 2 f (y f 2 (y 2 dy dy 2. R n R n Our aim is to obtain the compactness for the above commutators. Before stating our results, we briefly describe the background and our motivation. In [3], Calderón first proposed the concept of compactness in the multilinear setting and Bényi and Torres put forward an equivalent one in [2]. Bényi and Torres extended the result of compactness for linear singular integrals by Uchiyama [30] to the bilinear setting and obtained that [b, T ], [b, T ] 2, [b 2, [b, T ] ] 2 are compact bilinear operators from L p (R n L p2 (R n to L p (R n when b, b, b 2 CMO(R n, < p, p 2 < and /p + /p 2 = /p. Recently, Clop and Cruz [7] considered the compactness for the linear commutator on weighted spaces. For the bilinear case, Bényi et al. [] extended the result of [2] to the weighted case, and they obtained that all [b, T ], [b, T ] 2, [b 2, [b, T ] ] 2 are compact operators from L p (w L p2 (w 2 to L p (ν w when < p, p 2 <,

6 60 Appl. Math. J. Chinese Univ. Vol. 34, No. /p + /p 2 = /p <, w A p (R n A p (R n and b, b, b 2 CMO(R n. We note that in [], T is a Calderón-Zygmund operator with smooth kernel. Hence, in this article, we will consider the same compactness for these commutators by assuming T is an operator associated with non-smooth kernel. Although we will adopt the concept of compactness proposed in [2](The reader can refer to [2] and [3] for more properties of compact and precompact and some basic ideas used in [2,4,5,6,20,22,24,28], our proof meet some special difficulties so that some new ideas and estimates must be bought in. Particularly, some specific maximal functions will be involved. We denote the closed ball of radius r centered at the origin in the normed space X as B r,x = {x X : x r}. Definition.. A bilinear operator T : X Y Z is called compact if T (B,X B,Y is precompact in Z. Definition.2. A weight w belongs to the class A p (R n, < p <, if ( ( p sup w(ydy w(y dy p <. A weight w belongs to the class A (R n if there is a constant C such that w(ydy inf w(x. x Definition.3. Let p = (p, p 2 and /p = /p + /p 2 with p, p 2 <. Given w = (w, w 2, set ν w = 2 wp/pj j. We say that w satisfies the A p (R 2n condition if ( 2 ( /p sup ν w/p w p j j j <. ( Here, is understood as (inf w j,when p j =. /p j w p j j The following two theorems are our main results: Theorem.. Let T be a bilinear operator satisfying condition (7 and its kernel K satisfies (2, (8. Assume b CMO(R n, p, p 2 (,, p (, such that /p = /p + /p 2 and w = (w, w 2 A p (R 2n such that ν w A p (R n. Then [b, T ], [b, T ] 2 are compact from L p (w L p2 (w 2 to L p (ν w. In order to prove Theorem., we need the following result which has independent interest. Theorem.2. Let T be a bilinear operator satisfying condition (7 and its kernel K satisfies (2, (8. Assume b BMO(R n, p, p 2 (,, p (0, such that /p = /p + /p 2, w = (w, w 2 A p (R 2n. Then [b, T ] (f, f 2 Lp (ν w, [b, T ] 2 (f, f 2 Lp (ν w b BMO(Rn f L p (w f 2 L p 2 (w2. Remark.. Theorem., Theorem.2 are also true for the iterated commutator [b 2, [b, T ] ] 2, and their proofs are similar to the proofs of Theorem. and Theorem.2. We leave the details to the interested readers.

7 BU Rui, CHEN Jie-cheng. Compactness for the commutators of singular integral operators 6 We make some conventions. In this paper, we always denote a positive constant by C which is independent of the main parameters and its value may differ from line to line. We use the symbol A B to denote that there exists a positive constant C such that A B. For a measurable set E, χ E denotes its characteristic function. For a fixed p with p [,, p denotes the dual index of p. We also denote f = (f,, f m with scalar functions f j (j =, 2,..., m. Given α > 0 and a cube, l( denotes the side length of, and α denotes the cube which is the same center as and l(α = αl(. f denotes the average of f over. Let M be the standard Hardy-Littlewood maximal operator. For 0 < δ <, M δ is the maximal operator defined by ( /δ M δ f(x = M( f δ /δ (x = sup f(y dy δ, x M # is the sharp maximal operator defined by Fefferman and Stein [3], M # f(x = sup inf x c f(y c dy sup x f(y f dy, and M # δ f(x = M # ( f δ /δ (x. It is known that, when 0 < p, δ <, w A (R n, there exists a constant C > 0 such that (M δ f(x p w(xdx (M # δ f(xp w(xdx (9 R n R n for any function f for which the left-hand side is finite. 2 A multilinear maximal operator We need some basic facts about Orlicz spaces, for more information about these spaces the readers may consult [29]. For Φ(t = t( + log + t and a cube in R n, we define f L(log L, = inf{λ > 0 : It is obvious that f L(log L, > if and only if ( f(x Φ λ dx }. Φ( f(x dx >. The generalized Hölder inequality in Orlicz space together with the John-Nirenberg inequality imply that b(y b f(ydy b BMO(R n f L(log L,. Define the maximal operator M L(log L by M L(log L ( f(x 2 = sup f j L(log L,, x where the supremum is taken over all the cubes containing x. The following boundedness for M L(log L ( f was proved in [22]. Lemma 2.. If < p, p 2 <, p = 2 p j, and w = (w, w 2 A p (R 2n, then M L(log L ( f is bounded from L p (w L p2 (w 2 to L p (ν w.

8 62 Appl. Math. J. Chinese Univ. Vol. 34, No. Lemma 2. is helpful in the proof of Theorem.2. Besides this maximal operator, we need several other maximal operators in the following. In [22], a maximal function M( f was introduced, and its definition is M( f(x 2 ( = sup f j (y j dy j, x where the supremum is taken over all cubes containing x. The boundedness of M( f on weighted spaces was considered in [22, Theorem 3.3]. Furthermore, Grafakos, Liu, and Yang [4] introduced some new multilinear maximal operators: M 2, ( f(x ( ( = sup 2 kn f (y dy x k=0 2 k f 2 (y 2 dy 2, 2 k M 2,2 ( f(x ( ( = sup 2 kn f 2 (y 2 dy 2 2 k f (y dy, x k=0 where f = (f, f 2 and each f j (j {, 2} is a locally integrable function. The following boundedness of M 2, and M 2,2 were proved in [4]. 2 k Lemma 2.2. Let < p, p 2 <, p = 2 p j, and w = (w, w 2 A p (R 2n. Then M 2, and M 2,2 are bounded from L p (w L p2 (w 2 to L p (ν w. In addition, Hu [7] introduced another kind of bilinear maximal operators M β and M2 β which was defined by M β( f(x = sup f (y dy 2 kn 2 kβ x 2 k f 2 (y 2 dy 2, 2 k M 2 β( f(x = sup x f 2 (y 2 dy 2 2 kn 2 kβ 2 k 2 k f (y dy, where β R and the supremum is taken over all cubes containing x. As it is well known, a weight w A (R n implies that there exists a θ (0, such that for all cubes and any set E, For a fixed θ (0,, set w(e w( R θ = {w A (R n : w satisfies (0}. In [7], the following boundedness of M β and M2 β were proved. ( θ E. (0 Lemma 2.3. Let < p, p 2 <, p = 2 p j, w = (w, w 2 A p (R 2n and ν w R θ for some θ such that β < nθ min{/p, /p 2 }. Then M β and M2 β are bounded from Lp (w L p2 (w 2 to L p (ν w.

9 BU Rui, CHEN Jie-cheng. Compactness for the commutators of singular integral operators 63 3 Proof of Theorem.2 The proof of Theorem.2 will depend on some pointwise estimates using sharp maximal functions. The pointwise estimates are the following: Lemma 3.. Let T be a bilinear operator satisfying condition (7 and its kernel K satisfies (2, (8. If 0 < δ < 2, then for all f in any product of L pj (R n spaces with < p j <, M # δ (T ( f(x M( f(x 2 + C M 2,i ( f(x. The proof of this Lemma uses some ideas of [22, Theorem 3.2] and the following Lemma 3.2. Its proof is not difficult, so we omit. i= Lemma 3.2. Let T be a bilinear operator satisfying condition (7 and its kernel K satisfies (2, (8. If Tb, T b 2 be commutators with b BMO(Rn. For 0 < δ < ɛ with 0 < δ < /2, let r > and 0 < β < n. Then, there exists a constant C > 0, depending on δ and ɛ, such that 2 M # δ (T b( i f(x ( b BMO(Rn M L(log L ( f(x + M ɛ (T ( f(x 2 + {M i β(f r, f2 r (x} /r i= for all f = (f, f 2 of bounded functions with compact support. i= Proof. We only write out the estimate of M # δ (T b ( f(x, since the other can be obtained similarly. In our proof we will use some ideas of [28]. For a fixed x R n, a cube centered at x and constants c, λ, because 0 < δ < /2, inf c ( ( inf c ( C inf c ( C = I + I 2. T b ( f(z δ c δ dz /δ /δ Tb ( f(z c δ dz /δ (b(z λt (f, f 2 (z T ((b λf, f 2 (z c δ dz (b(z λt (f, f 2 (z δ dz /δ + inf c ( /δ C T ((b λf, f 2 (z c δ dz Let = 8 n, λ = b. The estimate of the first part I is the same as [28, Theorem 3.]. Therefore, we omit the proof, and from [28, Theorem 3.], we obtain that I b BMO(Rn M ɛ (T (f, f 2 (x. Now, we turn our attention to I 2. We decompose f, f 2 as f = f + f 2 = f (xχ + f (xχ R n \, f 2 = f 2 + f 2 2 = f 2 (xχ + f 2 (xχ R n \.

10 64 Appl. Math. J. Chinese Univ. Vol. 34, No. Let c = c + c 2 + c 3 and Therefore, c = T ((b λf, f 2 2 (x, c 2 = T ((b λf 2, f 2 (x, c 3 = T ((b λf 2, f 2 2 (x. ( /δ C I 2 T ((b λf, f2 (z δ dz ( /δ C + T ((b λf, f2 2 (z c δ dz ( /δ C + T ((b λf 2, f2 (z c 2 δ dz ( /δ C + T ((b λf 2, f2 2 (z c 3 δ dz = I 2 + I I I 4 2. We choose < q < /(2δ. By Hölder s inequality and the fact that T satisfies condition (7, we have ( /qδ C I2 T ((b λf, f2 (z qδ dz T ((b λf, f2 L /2, (, dz ( ( (b(z λf (z dz b BMO(Rn f L(log L, f 2 L(log L, b BMO(Rn M L(log L (f, f 2 (x. For I 2 2, by generalized Jensen s inequality, we get f2 (z dz T ((b λf, f2 2 (z T ((b λf, f2 2 (x K(z, y, y 2 (b λf (y f2 2 (y 2 dy dy 2 R 2n + K(x, y, y 2 (b λf (y f2 2 (y 2 dy dy 2 R 2n R ( z y 2n + z y 2 2n (b λf (y f2 2 (y 2 dy dy 2 + R ( x y 2n + x y 2 2n (b λf (y f2 2 (y 2 dy dy 2

11 BU Rui, CHEN Jie-cheng. Compactness for the commutators of singular integral operators 65 (b λf (y dy R n \ z y 2 2n f 2 2 (y 2 dy 2 + (b λf (y dy R n \ x y 2 2n f 2 2 (y 2 dy 2 (b λf (y dy 2 k \2 k z y 2 2n f 2 2 (y 2 dy 2 + (b λf (y dy 2 k \2 k x y 2 2n f 2 2 (y 2 dy 2 ( b BMO(R n f L(log L, 2 kn 2 k f2 2 (y 2 dy 2 2 k b BMO(Rn f L(log L, 2 kn f 2 L(log L,2 k b BMO(R n ( f (y r dy b BMO(Rn {M β(f r, f r 2 (x} r, r ( 2 kn 2 k f 2 (y 2 r dy 2 2 k where r >. Based on the above estimates, we deduce that For I 3 2, we obtain that I 2 2 b BMO(R n {M β(f r, f r 2 (x} r. T ((b λf 2, f2 (z T ((b λf 2, f2 (x R ( z y 2n + z y 2 2n (b λf 2 (y f2 (y 2 dy dy 2 + R ( x y 2n + x y 2 2n (b λf 2 (y f2 (y 2 dy dy 2 f2 (b λf 2 (y (y 2 dy 2 2 k \2 k z y 2n dy + f2 (b λf 2 (y (y 2 dy 2 2 k \2 k x y 2n dy f2 (y 2 dy 2 2 kn 2 k (b λf 2 (y dy 2 k + f2 (y 2 dy 2 f 2 (y 2 dy 2 2 kn 2 k b BMO(Rn f 2 L(log L, 2 kn 2 k (b b 2 k f 2 (y dy 2 k (b 2k b f 2 (y dy 2 k 2 kn f L(log L,2 k r

12 66 Appl. Math. J. Chinese Univ. Vol. 34, No. + b BMO(Rn f 2 L(log L, b BMO(R n + b BMO(Rn 2 kn k f L(log L,2 k ( f 2 (y 2 r dy 2 r ( f 2 (y 2 r dy 2 b BMO(R n {M 2 β(f r, f r 2 (x} r, where r >. This implies that r ( 2 kn 2 k f (y r dy 2 k ( 2 kn k 2 k f (y r dy 2 k r I 3 2 b BMO(Rn {M 2 β(f r, f r 2 (x} r. Finally, we use condition (8 to estimate I 4 2. Note that for any x, z and y, y 2 R n \, So, it is easy to verify that, x z nl( 8 min{ z y, z y 2 }. T ((b λf 2, f2 2 (z T ((b λf 2, f2 2 (x K(z, y, y 2 K(x, y, y 2 (b λf 2 (y f2 2 (y 2 dy dy 2 R 2n Rn\ l( γ R n \ ( z y + z y 2 2n+γ (b λf 2 (y f2 2 (y 2 dy dy 2 l( γ (b λf 2 (y 2 k l(< z y + z y 2 <2 k+ l( ( z y + z y 2 2n+γ f 2 2 (y 2 dy dy 2 l( γ ( ( (2 k l( 2n+γ (b λf 2 (y dy f2 2 (y 2 dy 2 2 k+2 2 k+2 ( ( 2 kγ 2 k+2 (b λf 2 (y dy 2 k+2 2 k+2 f2 2 (y 2 dy 2 2 k+2 b BMO(Rn M L(log L ( f(x. Therefore, I2 4 b BMO(R n M L(log L ( f(x. Combining the estimates for I2, I2 2, I2 3 and I2 4, lead to that I 2 b BMO(R n (M L(log L ( f(x 2 + {M i β(f r, f2 r (x} /r. The proof is completed. Now, we are ready to prove Theorem.2. Proof. For the sake of brevity, we only write out the proof of the boundedness of Tb, and the other can be got in the same method. By [22, Lemma 6.], we know that for every w A p (R 2n, there exists a finite constant < r 0 < min{p, p 2 } such that w A p/r0 (R 2n. From Lemma 2.3, for w A p/r0 (R 2n, there exists a β 0 > 0 satisfies that 2 i= Mi β 0 (f r0, f r0 2 (x is bounded i= r

13 BU Rui, CHEN Jie-cheng. Compactness for the commutators of singular integral operators 67 from L p/r0 (w L p2/r0 (w 2 to L p/r0 (ν w. Hence, 2 2 {M i β 0 (f r0, f r0 2 (x} r 0 L p (ν w = {M i β 0 (f r0, f r0 2 (x} /r0 L p/r 0 (ν w i= i= f r0 /r0 r0 f L p /r 0 (w 2 /r0 L p 2 /r 0 (w 2 = C f L p (w f 2 L p 2 (w2. Because ν w A 2p (R n A (R n, using inequality (9 and Lemma 3.2, we deduce that T b ( f Lp (ν w M δ (T b ( f Lp (ν w M # δ (T b ( f Lp (ν w b BMO(Rn M L(log L ( f(x + M ɛ (T ( f(x + b BMO(Rn + i= By Lemma 3., we have 2 {M i β 0 (f r0, f r0 2 (x} i= ( M L(log L ( f(x Lp (ν w + M ɛ (T ( f(x Lp (ν w 2 {M i β 0 (f r0, f r0 2 (x}/r0 L p (ν w. M ɛ (T ( f L p (ν w M # ɛ (T ( f L p (ν w M( f L p (ν w + C 2 M 2,i ( f(x L p (ν w i= M L(log L ( f Lp (ν w + C 2 M 2,i ( f(x Lp (ν w. i= r 0 Lp (ν w Now the desired result follows from Lemma 2., Lemma 2.2 and Lemma 2.3 directly. In the above proof, we note that when using the inequality (9 we need to explain that M ɛ (T ( f L p (ν w and M δ (T b ( f L p (ν w are finite. A detailed proof was given in page 33 of [22], and the proof can also be applied here owing to the boundedness of T which was given in [20, Theorem 2]. 4 Proof of Theorem. The idea of using truncated operators to prove compactness results in the linear setting can trace back to [2], and this method was adopted in [7]. Recently, Bényi et al. (see [] introduced a new smooth truncation to simplify the computations. We will use this technique to prove Theorem.. Let ϕ = ϕ(x, y, y 2 be a non-negative function in C c (R 3n, and it satisfy supp ϕ {(x, y, y 2 : max( x, y, y 2 < }, R 3n ϕ(udu =. For δ > 0, let χ δ = χ δ (x, y, y 2 be the characteristic function of the set {(x, y, y 2 : max( x

14 68 Appl. Math. J. Chinese Univ. Vol. 34, No. y, x y 2 3δ/2}, and let ψ δ = ϕ δ χ δ, where ϕ δ (x, y, y 2 = (δ/4 3n ϕ(4x/δ, 4y /δ, 4y 2 /δ. By an easy calculation, we get that ψ δ C (R 3n, ψ δ L (R 3n, supp ψ δ {(x, y, y 2 : max( x y, x y 2 δ}, and ψ δ (x, y, y 2 = if max( x y, x y 2 2δ. We define the truncated kernel K δ (x, y, y 2 = ψ δ (x, y, y 2 K(x, y, y 2, where K(x, y, y 2 is the kernel of the bilinear singular integral operator T considered in Theorem.. It s easy to verify that K δ also satisfies condition (2 and (8. Let T δ be the bilinear operator that associated with kernel K δ in the sense of (. The following Lemma was proved in []: Lemma 4.. For all x R n, b, b, b 2 C c lim [b, T δ ] [b, T ] L p (w L δ 0 p 2 (w 2 L p (ν w = 0, lim [b, T δ ] 2 [b, T ] 2 L p (w L δ 0 p 2 (w 2 L p (ν w = 0, (R n, if w = (w, w 2 A p (R 2n, then lim [b 2, [b, T δ ] ] 2 [b 2, [b, T ] ] 2 L p (w L δ 0 p 2 (w 2 L p (ν w = 0. By condition (2, Lemma 4. can be proved by the argument used in []. Lemma 4.2. Suppose that T is as in Theorem.. Then, for all ζ > 0, there exists a positive constant C such that for all f in the product of L pj (R n with < p j < and all x R n, T ( f(x ( M ζ (T ( f(x 2 + M 2,i ( f(x + M( f(x, where T ( f is the maximal truncated bilinear singular integral operator defined as T (f, f 2 (x = sup K(x, y, y 2 f (y f 2 (y 2 dy dy 2. η>0 i= max( x y, x y 2 >η The proof of the Lemma 4.2 is similar to the proof of [5, Theorem ], so we leave it to the interested reader. Lemma 4.3. Let < p <, w A p (R n and H L p (w. If (i H is bounded in L p (w; (ii lim A x >A f(x p w(xdx = 0 uniformly for f H; (iii lim t 0 f( + t f( L p (w = 0 uniformly for f H. Then H is precompact in L p (w.

15 BU Rui, CHEN Jie-cheng. Compactness for the commutators of singular integral operators 69 This Lemma was given in [7]. Now, we are ready to prove Theorem.. Proof. We will work with the commutator [b, T ] first, and the proof of the commutator [b, T ] 2 can be get similarly. From Lemma 4., we only need to prove the compactness for [b, T δ ] for any fixed δ /8. By Theorem.2, it suffices to show the result for b Cc (R n. Suppose f, f 2 belong to B (L p (w B (L p2 (w 2 = {(f, f 2 : f L p (w, f 2 L p 2 (w2 }, where w A p (R n. We need to prove that the following three conditions hold: (i [b, T δ ] (B (L p (w B (L p2 (w 2 is bounded in L p (ν w ; (ii lim A x >A [b, T δ ] (f, f 2 (x p ν w (xdx = 0; (iii Given 0 < ξ < /8, there exists a sufficiently small t 0 (t 0 = t 0 (ξ such that for all 0 < t < t 0, we have [b, T δ ] (f, f 2 ( [b, T δ ] (f, f 2 ( + t L p (ν w ξ. ( It is easy to find that the condition (i holds because of the boundedness of [b, T ] in Theorem.2 and Lemma 4.. Now, we prove the condition (ii using some ideas in [7]. Let R > 0 be large enough such that supp b B(0, R and let A max(2r,, l be a nonnegative integer. For any x > A, denote 2 VR(x 0 = K δ (x, y, y 2 f j (y j dy dy 2, and VR(x l = y 2 x 2 l x y 2 2 l x y R y R K δ (x, y, y 2 2 f j (y j dy dy 2, where l > 0. From condition (2, we deduce that VR(x l 2 l x y 2 2 l x y R ( x y + x y 2 2n f (y f 2 (y 2 dy dy 2 f (y f 2 (y 2 2 l x y 2 2 l x y R ( x + x y 2 2n dy dy 2 (2 l x 2n f (y f 2 (y 2 dy dy 2 2 l x y 2 2 l x y R ( (2 l x 2n w ( p p (y dy w p 2 p 2 2 (y 2 dy 2. B(0,R The same estimate can be got for VR 0 constant θ (0, such that B(0,R w p (y dy (2 (j+l RA nθ B(0,2 l x (x. Note that w p A (R n, so there exists a B(0,2 l+j A w p (y dy.

16 70 Appl. Math. J. Chinese Univ. Vol. 34, No. Since p >, it follows that ( /p [b, T δ ] (f, f 2 (x p ν w (xdx 2 j A x 2 j A ( /p VR(x l p ν w (xdx l=0 2 j A x 2 j A ( /p (2 l x 2np ν w(xdx Thus, l=0 ( B(0,R 2 j A x 2 j A w p (y dy /p ( B(0,2 l+j A ( (2 l+j 2 A 2n (2 (j+l RA nθ( /p l=0 ( as A. B(0,2 l+j A w p (y dy /p ( B(0,2 l+j A w p 2 2 (y 2 dy 2 /p 2 B(0,2 j A (2 l+j A 2n (2 (j+l RA nθ( /p (2 j+l A 2n l=0 2 l( nθ( /p 2 j( nθ( /p (R/A nθ( /p l=0 2 j( nθ( /p (R/A nθ( /p. /p ν w (xdx w p 2 2 (y 2 dy 2 /p 2 ( /p [b, T δ ] (f, f 2 (x p ν w (xdx (R/A nθ( /p 0, x >A So, it suffices to verify condition (iii. We denote E = {(x, y, y 2 : min( x y, x y 2 > η}, F = {(x, y, y 2 : max( x y, x y 2 > 2δ}, G = {(x, y, y 2 : max( x y, x y 2 > η}, H = {(x, y, y 2 : δ < max( x y, x y 2 < 2δ}. To prove (, we decompose the expression inside the L p (ν w norm as follows: [b, T δ ] (f, f 2 (x [b, T δ ] (f, f 2 (x + t 2 = K δ (x, y, y 2 (b(x b(x + t f j (y j dy dy 2 E + (K δ (x, y, y 2 K δ (x + t, y, y 2 (b(x + t b(y E 2 + K δ (x, y, y 2 (b(x b(y f j (y j dy dy 2 E c 2 f j (y j dy dy 2

17 BU Rui, CHEN Jie-cheng. Compactness for the commutators of singular integral operators 7 + K δ (x + t, y, y 2 (b(y b(x + t E c = A(x + B(x + C(x + D(x, 2 f j (y j dy dy 2 where 0 < η < and the choice of η will be specified later. It is obvious that K δ (x, y, y 2 = K(x, y, y 2 on F. Consequently, K δ (x, y, y 2 f (y f 2 (y 2 dy dy 2 K(x, y, y 2 f (y f 2 (y 2 dy dy 2 E G = K δ (x, y, y 2 f (y f 2 (y 2 dy dy 2 (E F (E H K(x, y, y 2 f (y f 2 (y 2 dy dy 2 (E F (G\(E F K δ (x, y, y 2 f (y f 2 (y 2 dy dy 2 E H + K(x, y, y 2 f (y f 2 (y 2 dy dy 2 G E c + K(x, y, y 2 f (y f 2 (y 2 dy dy 2 G F c E + K(x, y, y 2 f (y f 2 (y 2 dy dy 2. G F c E c Now, we estimate the above four parts. From condition (2, we have K δ f (y f 2 (y 2 (x, y, y 2 f (y f 2 (y 2 dy dy 2 ( x y + x y 2 2n dy dy 2 and E H H M(f, f 2 (x, K(x, y, y 2 f (y f 2 (y 2 dy dy 2 G E c f (y f 2 (y 2 x y <η x y 2 >η ( x y + x y 2 2n dy dy 2 f 2 (y 2 f (y dy x y 2 2n dy 2 x y <η 2 kn B(x, η 2 M 2,i (f, f 2 (x, i= B(x,η 2 k η< x y 2 <2 k η f (y dy B(x, 2 k η B(x,2 k η f 2 (y 2 dy 2 where the set G E c includes {(x, y, y 2 : x y < η, x y 2 > η} and {(x, y, y 2 : x y > η, x y 2 < η}. Since the estimates on these two regions are similar, we omit the latter. This method will be used several times in the following.

18 72 Appl. Math. J. Chinese Univ. Vol. 34, No. and Because η < x y < 2δ, η < x y 2 < 2δ when (x, y, y 2 G F c E. Hence, f (y f 2 (y 2 K(x, y, y 2 f (y f 2 (y 2 d y 4δ d y ( x y + x y 2 2n+ G F c E K(x, y, y 2 f (y f 2 (y 2 d y G F c E c G δ η M(f, f 2 (x, x y <η x y 2 >η 2 M 2,i (f, f 2 (x. i= f (y f 2 (y 2 d y ( x y + x y 2 2n In summary, we get A(x t b L (R n K δ (x, y, y 2 f (y f 2 (y 2 d y E t K δ (x, y, y 2 f (y f 2 (y 2 d y K(x, y, y 2 f (y f 2 (y 2 d y E + C t T (f, f 2 (x ( t T (f, f 2 (x + η M(f, f 2 (x + 2 i= G M 2,i (f, f 2 (x. This, along with Lemma 2.2, Lemma 3., Lemma 4.2 and [22, Theorem 3.7], leads to that A(x Lp (ν w t ( + /η. (2 In order to estimate B(x, by a consequence of condition (8, we have K(x, y, y 2 K(x D x x γ, y, y 2 ( x y + x y 2 2n+γ when x x 8 min{ x y, x y 2 }. Then B(x b L (R n K δ (x, y, y 2 K δ (x + t, y, y 2 f (y f 2 (y 2 d y E t γ f (y f 2 (y 2 d y ( x y + x y 2 2n+γ Therefore, G t γ η γ M(f, f 2 (x. B(x Lp (ν w t γ η γ. (3 For any 0 < β <, we have b(x b(y x y β. Thus, from the size condition (2 and the property of the support of K δ (x, y, y 2, we can estimate C(x:

19 BU Rui, CHEN Jie-cheng. Compactness for the commutators of singular integral operators 73 C(x b L (R n η x y <η + C x y >η ηm 2, ( f(x + C x y 2 <η x y 2 <η x y 2 >η ηm 2, ( f(x + Cη β M 2 β( f(x, f (y f 2 (y 2 ( x y + x y 2 2n dy dy 2 f (y f 2 (y 2 ( x y + x y 2 2n β dy dy 2 f 2 (y 2 dy 2 2 k η< x y <2 k η provided η < δ. From Lemma 2.2 and Lemma 2.3, we deduce that when we take sufficiently small β. f (y x y 2n β dy C(x Lp (ν w η, (4 Finally, for the last part D(x we proceed in a similar way, by replacing x with x+t and the region of integration E c with a larger one {(x, y, y 2 : min( x + t y, x + t y 2 < η + t }. By the fact that x B(x + t, η + t, where B(x + t, η + t denote the ball centered at x + t and with radius η + t, we obtain D(x Lp (ν w ( t + η. (5 Now, let us define t 0 = ξ 2 and for each 0 < t < t 0, choose η = t /ξ. Then inequalities (2-(5 imply (, and in this way, we can conclude that [b, T ] is compact. By symmetry, [b, T ] 2 is also compact. References [] Á Bényi, W Damián, K Moen, R H Torres. Compact bilinear commutators: the weighted case, Michigan Math J, 205, 64: [2] Á Bényi, R H Torres. Compact bilinear operators and commutators, Proc Amer Math Soc, 203, 4: [3] A P Calderón. Intermediate spaces and interpolation, the complex method, Studia Math, 964, 24: [4] J C Chen, G E Hu. Compact commutators of rough singular integral operators, Canad Math Bull, 205, 58: [5] Y P Chen, Y Ding, X X Wang. Compactness of commutators for singular integrals on Morrey s- paces, Canad J Math, 202, 64: [6] Y P Chen, Y Ding, X X Wang. Compactness of commutators of Riesz potential on Morrey spaces, Potential Anal, 2009, 30:

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21 BU Rui, CHEN Jie-cheng. Compactness for the commutators of singular integral operators 75 [25] F Liu, H X Wu. Endpoint regularity of multisublinear fractional maximal functions, Canad Math Bull, 207, 60: [26] F Liu, Y Xue, K Yabuta. Rough singular integral and maximal operators supported by subvarieties on Triebel-Lizorkin spaces, Nonlinear Anal, 208, 7: [27] C Pérez. Endpoint estimates for commutators of singular integral operators, J Funct Anal, 995, 28: [28] C Pérez, G Pradolini, R H Torres, R Trujillo-González. End-points estimates for iterated commutators of multilinear singular integrals, Bull London Math Soc, 204, 46: [29] M M Rao, Z D Ren. Theory of Orlicz Space, Marcel Dekker, New York, 99. [30] A Uchiyama. On the compactness of operators of Hankel type, Tôhoku Math J, 978, 30: [3] K Yosida. Functional Analysis, Springer-Verlag, Berlin, 995. [32] X Yu, J C Chen, Y D Zhang. A weight estimates for vector-valued commutators of multilinear fractional integral, Appl Math J Chinese Univ Ser B, 203, 28(3: Department of Mathematics, ingdao University of Science and Technology, ingdao 26606, China burui0@63.com 2 Department of Mathematics, Zhejiang Normal University, Jinhua 32004, China jcchen@zjnu.edu.cn