A characterization of the uniform strong type $(1,1)$ bounds for averaging operators

Abstract

We prove that in a metric measure space $(X,d,\mu )$, the averaging operators $A_{r,\mu }$ satisfy a uniform strong type $(1,1)$ bound ${\sup }_{r,\mu }\Vert A_{r,\mu }{\Vert }_{L^1\to L^1}<\infty$ if and only if $X$ satisfies a certain geometric condition, the equal radius Besicovitch intersection property.

arXiv admin note: text overlap with arXiv:1811.10478