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<p>We obtain a necessary and sufficient condition on an exponent <i>p</i>(·) for which the Hardy–Littlewood maximal operator is bounded on the variable <i>L</i><sup><i>p</i>(·)</sup> space. It is formulated in terms of the Muckenhoupt-type condition <i>A</i><sub><i>p</i>(·)</sub>, responsible for a local control of <i>p</i>(·), and a certain integral condition on <i>p</i>(·), responsible for the behaviour of <i>p</i>(·) at infinity. Our approach is based on an earlier characterization established by L. Diening and on non-increasing rearrangements.</p>

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A boundedness criterion for the maximal operator on variable Lebesgue spaces

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  • 发表日期：

    2025年7月13日

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  • Andrei K. Lerner

摘要

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