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

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

8 pages