Boundedness of Sublinear Operators Generated by Calderón-Zygmund Operators on Generalized Weighted Morrey Spaces

Abstract

In this paper we study the boundedness for a large class of sublinear operators T generated by Calderón-Zygmund operators on generalized weighted Morrey spaces Mp,φ(w) with the weight function w(x) belonging to Muckenhoupt's class Ap. We find the sufficient conditions on the pair (φ1, φ2) which ensures the boundedness of the operator T from one generalized weighted Morrey space Mp,φ 1 (w) to another Mp,φ 2 (w) for p > 1 and from M1,φ 1 (w) to the weak space W M1,φ 2 (w). In all cases the conditions for the boundedness are given in terms of Zygmund-type integral inequalities on (φ1, φ2), which do not assume any assumption on monotonicity of φ1, φ2 in r. Conditions of these theorems are satisfied by many important operators in analysis, in particular pseudodifferential operators, Littlewood-Paley operator, Marcinkiewicz operator and Bochner-Riesz operator.

Multilinear commutators of vector-valued intrinsic square functions on vector-valued generalized weighted Morrey spaces

“…Recently, Komori and Shirai [29] first defined the weighted Morrey spaces L p,κ (w) and studied the boundedness of some classical operators such as the Hardy-Littlewood maximal operator, the Calderón-Zygmund operator on these spaces. Also, Guliyev [21,22] introduced the generalized weighted Morrey spaces M p,ϕ w and studied the boundedness of the sublinear operators and their higher order commutators generated by Calderón-Zygmund operators and Riesz potentials in these spaces (see, also [25,27,28,35]).…”

Fractional multilinear integrals with rough kernels on generalized weighted Morrey spaces

“…Recently, Komori and Shirai [34] considered the weighted Morrey spaces L p;Ä .w/ and studied the boundedness of some classical operators such as the Hardy-Littlewood maximal operator, the Calderón-Zygmund operator on these spaces. Also, Guliyev [24] introduced the generalized weighted Morrey spaces M p;' .w/ and studied the boundedness of the classical operators and its commutators in these spaces M p;' .w/, see also [24,29,33,40]. In [24] the author gave a concept of generalized weighted Morrey space M p;' .w/ which could be viewed as extension of both generalized Morrey space M p;' and weighted Morrey space…”

Weighted local Morrey spaces

“…Many researchers investigated the boundedness properties of the linear operators acting on weighted Morrey spaces. such as sublinear operator [4,12,15,35], singular integral operators [14,35,63], commutators [17,12,35,59,61], pseudo-differential operators [26], the square functions [11], Toeplitz operators [56], the fractional integral operators [12,28,31,32] and fractional integrals associated to operators [51,54,55] including the related commutators. Applications to partial differential equations can be found in [8,19,50].…”