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arXiv · 2302.02475

A boundedness criterion for the maximal operator on variable Lebesgue spaces

Andrei K. Lerner
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Abstract

We obtain a necessary and sufficient condition on an exponent $p(\cdot)$ for which the Hardy--Littlewood maximal operator is bounded on the variable $L^{p(\cdot)}$ space. It is formulated in terms of the Muckenhoupt-type condition $A_{p(\cdot)}$, responsible for a local control of $p(\cdot)$, and a certain integral condition on $p(\cdot)$, responsible for the behaviour of $p(\cdot)$ at infinity. Our approach is based on an earlier characterization established by L. Diening and on non-increasing rearrangements.

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BibTeXRIS

Andrei K. Lerner. 2023-02-05. A boundedness criterion for the maximal operator on variable Lebesgue spaces. https://arxiv.org/abs/2302.02475

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