The maximal operator on variable Lebesgue spaces: an ${\mathcal A}_{\infty}$-characterization

TL;DR

This paper introduces a new criterion for the boundedness of the maximal operator on variable Lebesgue spaces, based on an analogue of the weighted $A_{ ext{infty}}$ condition, advancing understanding in harmonic analysis.

Contribution

It provides a novel boundedness criterion for the maximal operator on variable Lebesgue spaces using an $A_{ ext{infty}}$-type condition, extending classical weighted theory.

Findings

New boundedness criterion for $M$ on $L^{p( ext{·})}$

Characterization in terms of variable $A_{ ext{infty}}$ condition

Enhanced understanding of maximal operator behavior in variable spaces

Abstract

In this paper we obtain a new boundedness criterion for the maximal operator $M$ on variable exponent spaces $L^{p(\cdot)}$. It is formulated in terms of the variable exponent analogue of the well known weighted $A_{\infty}$ condition.