Endpoint estimates for the commutators of multilinear Calderón-Zygmund operators with Dini type kernels

Li and Xue Journal of Inequalities and Applications (2016) 2016:252 DOI 10.1186/s13660-016-1201-2 RESEARCH Open Access Endpoint estimates for the commutators of multilinear Calderón-Zygmund operators with Dini type kernels Zhengyang Li and Qingying Xue* * Correspondence: School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing Normal University, Beijing, 100875, People’s Republic of China Abstract Let Tb and Tb be the commutators in the jth entry and iterated commutators of the multilinear Calderón-Zygmund operators, respectively. It was well known that the commutators of linear Calderón-Zygmund operators were not of weak type (1, 1) and (H1 , L1 ), but they did satisfy certain endpoint L log L type estimates. In this paper, our aim is to give more natural sharp endpoint results. We show that Tb and Tb are 1 bounded from the product Hardy space H1 × · · · × H1 to weak L m ,∞ space, whenever the kernel satisfies a class of Dini type condition. This was done by using a key lemma given by Christ, a very complex decomposition of the integrand domains, and carefully splitting the commutators into several terms. Keywords: commutators; multilinear Calderón-Zygmund operator; C-Z kernel of ω type; Dini type conditions; Hardy spaces 1 Introduction 1.1 Commutators of classical C-Z operators In , Coifman, Rochberg, and Weiss [] first introduced and studied the commutator of classical linear Calderón-Zygmund singular integrals, which was defined by Tb f = [b, T]f = bT(f ) – T(bf ). The Lp boundedness of Tb was given in [] for  < p < ∞ when b ∈ BMO(Rn ). It is well known that Tb fails to be of weak type (, ) and is not bounded from H  (Rn ) to L (Rn ). Counterexamples were given by Pérez [] and Paluszyński []. As an alternative result of the weak (, ) estimate of Tb , Pérez [] obtained the following L(log L) type endpoint estimate:      x ∈ Rn : Tb f (x) > λ  ≤ C     |f (x)| |f (x)|  + log+ dx, λ λ Rn λ > . Moreover, alternative results of the (H  , L ) boundedness were also considered in the work of Alvarez [], Pérez [], and Liang, Ky, and Yang [], which concerned with the boundedness of Tb on the subspace of atomic Hardy spaces, or concerned with the (Hw , Lw ) © 2016 Li and Xue. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Li and Xue Journal of Inequalities and Applications (2016) 2016:252 Page 2 of 22 boundedness of Tb if b belongs to a subspace of BMO which is associated to the weight function w. On the other hand, another more reasonable and alternative result of weak type (, ) and (H  , L ) estimate was given by Liu and Lu [] in . The authors [] showed that Tb is bounded from H  (Rn ) to L,∞ (Rn ) if b ∈ BMO(Rn ). We note that Tb also fails to be bounded from H p (Rn ) to Lp,∞ (Rn ) for  < p <  by the generalized interpolation theorem [], pp.. Therefore, the (H  , L,∞ ) boundedness of Tb becomes a sharp endpoint estimate. Moreover, always L(log L)(Sn– )  H  (Sn– ) if f vanishes on the unit sphere. However, there is no such inclusion relationship on Rn . Moreover, the inverse including relationship is still not true, since the following example shows that H  (Rn )  L(log L)(Rn ). Example . Let f (x) = χ[–  ,  ]   aj (x) = Thus, f (x) = f (x) {χ   + χ[  ,  ] } × j ,  f ( j+ ) [– j ,– j+ ] j+ j λj =  ) f ( j+ j . ∞ ∞ j= λj aj (x), and it is easy to verify that each aj is a (, ∞, )-atom. Notice that ∞ |λj | = j= for some ε > ,  x log+ε  |x| j=  |f ( j+ )| j ∞ ≤ ∞    ·  = < ∞, +ε j+ j  j+ log  (j + )+ε j= j= then we have f ∈ H  (Rn ). Obviously, f ∈/ L(log L)(Rn ). Thus, the (H  , L,∞ ) boundedness and the L log L type estimate of Tb are independent in the sense that one cannot cover the results of the other. 1.2 Commutators of multilinear operators In recent years, the theory of multilinear Calderón-Zygmund operators with standard kernels have been developed very quickly and a lot of work has been done. Among such achievements is the celebrated work of Coifman and Meyer [–], Christ and Journé [], Kenig and Stein [], Grafakos and Torres [, ], and Lerner et al. []. In order to state some well-known results, we need to introduce some definitions. Definition . (C-Z kernel of ω type [, ]) Let ω(t) be a non-negative and nondecreasing function on R+ . Let K(x, y , . . . , ym ) be a locally integrable function defined away from the diagonal x = y = · · · = ym in (Rn )m+ . Denote (x, y) = (x, y , . . . , ym ), we say K is an m-linear Calderón-Zygmund kernel of ω type, if there exists a positive constant C such that   K(x, y) ≤ m C , ( j= |x – yj |)mn     |x – x | K(x, y) – K x , y  ≤ m C  , ω m ( j= |x – yj |)mn j= |x – yj | (.) (.) Li and Xue Journal of Inequalities and Applications (2016) 2016:252 Page 3 of 22 whenever |x – x | ≤  max≤j≤m |x – yj |, and   K(x, y , . . . , yi , . . . , ym ) – K x, y , . . . , y , . . . , ym  i   C |yi – yi | , ≤ m ω m ( j= |x – yj |)mn j= |x – yj | (.) whenever |yi – yi | ≤  max≤j≤m |x – yj |. Definition . (Multilinear C-Z singular integral operators [, ]) Let K(x, y) be a CZ kernel of ω type. For any f = (f , . . . , fm ) ∈ S (Rn ) × S (Rn ) × · · · × S (Rn ) and all x ∈/ m j= supp fj , we define the multilinear Calderón-Zygmund singular integral operators as follows:  K(x, y , . . . , ym )f (y ), . . . , fm (ym ) dy · · · dym . T(f)(x) = (Rn )m Definition . (Commutators of multilinear C-Z operators) Let bj ∈ BMO(Rn ) and T be the operator defined in Definition .. The commutators in the jth entry and the iterated commutators of T are defined by Tb (f)(x) = m Tb (f)(x) j j= m bj (x)T(f , . . . , fj , . . . , fm )(x) – T(f , . . . , bj fj , . . . , fm )(x) =  (.) j= and    Tb (f) = b , b , . . . bm– , [bm , T]m , m– · · ·   (f)  m  = bj (x) – bj (yj ) K(x, y , . . . , ym )f (y ) · · · fm (ym ) dy. (.) (Rn )m j= Remark . Obviously, in the special case, ω(t) = t ε for some ε > , then the operator T defined in Definition . coincides with the standard multilinear Calderón-Zygmund operator defined and studied by Grafakos and Torres []. Moreover, if ω(t) = t ε , the weighted  j  strong and L(log L) type endpoint estimates for Tb (f , . . . , fm )(x) = m j= Tb (f ) and Tb have already been studied in [] and [], respectively. Definition . (Dini(a) type conditions) Let ω(t) be a non-negative and non-decreasing function on R+ . ω is said to satisfy the Dini( (...truncated)