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Characterization of rearrangement invariant spaces with fixed points for the Hardy-Littlewood maximal operator

Joaquim Martin·Javier Soria
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Abstract

We characterize the rearrangement invariant spaces for which there exists a non-constant fixed point, for the Hardy-Littlewood maximal operator (the case for the spaces $L^p(\mathbb{R}^{n})$ was first considered by Korry in \cite{Ko}). The main result that we prove is that the space $L^{\frac{n}{n-2},\infty}(\mathbb{R}^{n})\cap L^{\infty}(\mathbb{R}^{n})$ is minimal among those having this property

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Joaquim Martin, Javier Soria. 2004-09-01. Characterization of rearrangement invariant spaces with fixed points for the Hardy-Littlewood maximal operator. https://arxiv.org/abs/math/0409017

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