Maximal, potential and singular operators in the local "complementary" variable exponent Morrey type spaces

TL;DR

This paper studies local 'complementary' Morrey spaces with variable exponents, establishing boundedness of key operators like maximal, singular, and potential operators without requiring monotonicity of the defining function.

Contribution

It introduces and analyzes new local Morrey spaces with variable exponents, proving boundedness of classical operators under Zygmund-type conditions without monotonicity assumptions.

Findings

Boundedness of Hardy-Littlewood maximal operator in these spaces

Boundedness of Calderon-Zygmund singular integral operators

Sobolev-type embedding theorems for potential operators

Abstract

We consider local "complementary" generalized Morrey spaces ${\dual \cal M}_{\{x_0\}}^{p(\cdot),\om}(\Om)$ in which the $p$-means of function are controlled over $\Om\backslash B(x_0,r)$ instead of $B(x_0,r)$, where $\Om \subset \Rn$ is a bounded open set, $p(x)$ is a variable exponent, and no monotonicity type conditio is imposed onto the function $\om(r)$ defining the "complementary" Morrey-type norm. In the case where $\om$ is a power function, we reveal the relation of these spaces to weighted Lebesgue spaces. In the general case we prove the boundedness of the Hardy-Littlewood maximal operator and Calderon-Zygmund singular operators with standard kernel, in such spaces. We also prove a Sobolev type ${\dual \cal M}_{\{x_0\}}^{p(\cdot),\om} (\Om)\rightarrow {\dual \cal M}_{\{x_0\}}^{q(\cdot),\om} (\Om)$-theorem for the potential operators $I^{\al(\cdot)},$ also of variable order. In all the cases the conditions for the boundedness are given it terms of Zygmund-type integral inequalities on $\om(r)$, which do not assume any assumption on monotonicity of $\om(r)$.