Boundedness of averaging operators on geometrically doubling metric spaces

TL;DR

This paper proves that averaging operators are uniformly bounded on $L^1$ spaces across all geometrically doubling metric measure spaces, ensuring measure-independent bounds and $L^1$ convergence of averages as the radius approaches zero.

Contribution

It establishes the measure-independent boundedness of averaging operators on $L^1$ in geometrically doubling metric spaces, a significant generalization.

Findings

Averaging operators are uniformly bounded on $L^1$

Bounds are independent of the measure

Ensures $L^1$ convergence of averages as $r o 0$

Abstract

We prove that averaging operators are uniformly bounded on $L^1$ for all geometrically doubling metric measure spaces, with bounds independent of the measure. From this result, the $L^1$ convergence of averages as $r \to 0$ immediately follows.