Estimates of singular integrals and multilinear commutators in weighted Morrey spaces

Xiao Feng Ye,Xu Sheng Zhu

doi:10.1186/1029-242x-2012-302

Xiao Feng Ye, Xu Sheng Zhu

Open Access

https://doi.org/10.1186/1029-242x-2012-302

Copy DOI

Abstract

Suppose T is a singular integral operator whose kernel is a variable kernel with mixed homogeneity; the purpose of this paper is to study the continuity of the operator in weighted Morrey spaces L p,κ (ω), 1 ≤ p < ∞ ,0< κ < 1. A special attention is paid also to the multilinear commutator of this operator with BMO function. MSC: 42B20; 42B35

Highlights

Let K(x, ξ ) : Rn × Rn\{ } → R be a variable kernel
The variable kernel K(x, ξ ) depends on some parameter x and possesses ‘good’ properties with respect to the second variable ξ, which was firstly introduced by Fabes and Rieviève in [ ]
The weighted Morrey spaces Lp,κ (w) were introduced by Komori and Shirai [ ]. They showed some classical integral operators and corresponding commutators were bounded in weighted Morrey spaces

Summary

Introduction

Let K(x, ξ ) : Rn × Rn\{ } → R be a variable kernel. The singular integral operator is defined byTf (x) = p.v. They showed some classical integral operators and corresponding commutators were bounded in weighted Morrey spaces. The main purpose of this paper is to discuss the continuity of the singular integral operator whose kernel is a variable kernel with mixed homogeneity and its multilinear commutator in the weighted Morrey spaces Lp,κ (ω), < p < ∞, < κ < , where the weight function ω is Ap weight.

Objectives

Results

Conclusion