Weighted Boundedness of the Maximal, Singular and Potential Operators in Variable Exponent Spaces

TL;DR

This paper surveys recent advances in the boundedness of classical operators like the maximal and singular operators within weighted variable exponent spaces, introducing new results especially for metric measure spaces with doubling measures.

Contribution

It provides new boundedness results for the Hardy-Littlewood maximal operator in variable exponent spaces over metric measure spaces with doubling measures, including novel weight classes.

Findings

Boundedness of the Hardy-Littlewood maximal operator in variable exponent spaces.

Introduction of new weight classes with specific growth conditions.

Some results are new even for constant exponent cases.

Abstract

We present a brief survey of recent results on boundedness of some classical operators within the frameworks of weighted spaces $L^{p(\cdot)}(\varrho)$ with variable exponent $p(x)$, mainly in the Euclidean setting and dwell on a new result of the boundedness of the Hardy-Littlewood maximal operator in the space $L^{p(\cdot)}(X,\varrho)$ over a metric measure space $X$ satisfying the doubling condition. In the case where $X$ is bounded, the weight function satisfies a certain version of a general Muckenhoupt-type condition For a bounded or unbounded $X$ we also consider a class of weights of the form $\varrho(x)=[1+d(x_0,x)]^{\bt_\infty}\prod_{k=1}^m w_k(d(x,x_k))$, $x_k\in X$, where the functions $w_k(r)$ have finite upper and lower indices $m(w_k)$ and $M(w_k)$. Some of the results are new even in the case of constant $p$.