Boundedness for Sublinear Operators with Rough Kernels on Weighted Grand Morrey Spaces

Junmei Wang

doi:10.11648/j.pamj.20190801.13

Abstract

In this paper, we study the boundedness of some sublinear operators with rough kernels, satisfied by most of the operators in classical harmonic analysis, on the generalized weighted grand Morrey spaces. More specifically, we show that the sublinear operators with rough kernels are bounded on these spaces under the conditions that the operators and the kernel functions satisfy some size conditions, and the operators are bounded on Lebesgue spaces. This is done by exploiting the well-known boundedness of sublinear operators with rough kernels on Lebesgue spaces, a more explicit decomposition of the generalized weighted grand Morrey spaces and the good properties of the weight functions and the kernel functions. Through combining some properties of Ap weight with the relevant lemmas of operators with rough kernel, we obtain the boundedness for sublinear operators with rough kernels on weighted grand morrey spaces. Furthermore, using the equivalent norm and the properties of BMO functions, an application of the boundedness of the sublinear operators with rough kernels to the corresponding commutators formed by certain operators and BMO functions are also considered. And the boundedness of commutator is obtained by the lemma of function BMO.

Highlights

Morrey [1] first introduced the classical Morrey spaces to investigate the local behavior of solutions to second order elliptic partial differential equations (PDE)
We study some sublinear operators with rough kernels on generalized weighted grand Morrey spaces
Boundedness of Sublinear Operators The topic of this paper is intended as an attempt to study the boundedness of some sublinear operators with rough kernels which satisfy (4) and (5) on generalized weighted grand Morrey spaces and give some criterions to deduce the boundedness of the sublinear operators on certain spaces

Summary

Introduction

Morrey [1] first introduced the classical Morrey spaces to investigate the local behavior of solutions to second order elliptic partial differential equations (PDE). Inspired by the works of [26] and [19], we consider some size conditions (the following (4) and (5)) which are more general than (3) on the generalized weighted grand Morrey spaces. Zheng, Zhang and Shi [29] introduced the boundedness for sublinear operators on generalized weighted grand Morrey spaces. They studied the boundedness of some sublinear operators, satisfied by most of the operators in classical harmonic analysis, on the generalized weighted grand Morrey spaces They considered the applications to the corresponding commutators formed by certain operators and bounded mean oscillations ( BMO ) functions. We study some sublinear operators with rough kernels on generalized weighted grand Morrey spaces.

Method and Result

Boundedness of Commutators

Conclusion