On a dual property of the maximal operator on weighted variable $L^p$ spaces

TL;DR

This paper extends a known dual property of the maximal operator from unweighted to weighted variable Lebesgue spaces, showing that boundedness on one space implies boundedness on its dual.

Contribution

The paper generalizes Diening's dual property of the maximal operator to weighted variable Lebesgue spaces, broadening its applicability.

Findings

Boundedness of M on L^{p( abla)} implies boundedness on L^{p'( abla)}

Extension to weighted spaces broadens the duality understanding

Supports further analysis in weighted variable Lebesgue spaces

Abstract

L. Diening \cite{D1} obtained the following dual property of the maximal operator $M$ on variable Lebesque spaces $L^{p(\cdot)}$: if $M$ is bounded on $L^{p(\cdot)}$, then $M$ is bounded on $L^{p'(\cdot)}$. We extend this result to weighted variable Lebesque spaces.