Multilinear Commutators of Calderón-Zygmund Operator on Generalized Weighted Morrey Spaces

The boundedness of multilinear commutators of Calderón-Zygmund operator on generalized weighted Morrey spaces with the weight function belonging to Muckenhoupt's class is studied. When and , , , the sufficient conditions on the pair which ensure the boundedness of the operator from to are found. In all cases the conditions for the boundedness of are given in terms of Zygmund-type integral inequalities on , which do not assume any assumption on monotonicity of in . 1. Introduction Let be a Calderón-Zygmund singular integral operator and . A well known result of Coifman et al. [1] states that if and is a Calderón-Zygmund operator, then the commutator operator is bounded on for . The commutators of Calderón-Zygmund operator play an important role in studying the regularity of solutions of elliptic, parabolic and ultraparabolic partial differential equations of second order (see, [2–7]). The classical Morrey spaces were originally introduced by Morrey in [8] to study the local behavior of solutions to second order elliptic partial differential equations. For the properties and applications of classical Morrey spaces, we refer the readers to [2–4, 8, 9]. Let , and be locally integrable functions when we consider multilinear commutators as defined by where is Calderón-Zygmund kernel. That is, for all distinct , and all with , there exist positive constant and such that(i) (ii) ; and(iii) when , it is the classical commutator which was introduced by Coifman et al. in [1]. It is well known that Calderón-Zygmund operators play an important role in harmonic analysis (see [10–12]). We define the generalized weighed Morrey spaces as follows. Definition 1. Let , be a positive measurable function on and be non-negative measurable function on . We denote by the generalized weighted Morrey space, the space of all functions with finite norm where denotes the weighted -space of measurable functions for which Furthermore, by we denote the weak generalized weighted Morrey space of all functions for which where denotes the weak -space of measurable functions for which Remark 2. (1) If , then is the generalized Morrey space. (2) If , then is the weighted Morrey space. (3) If , then is the two weighted Morrey space. (4) If and with , then is the classical Morrey space and is the weak Morrey space. (5) If , then is the weighted Lebesgue space. The commutators are useful in many nondivergence elliptic equations with discontinuous coefficients, [2–5]. In the recent development of commutators, Pérez and Trujillo-González [13] generalized these multilinear commutators and proved