Compactness of averaging operators on non-reflexive Lebesgue spaces

TL;DR

This paper characterizes when averaging operators on non-reflexive Lebesgue spaces over certain metric measure spaces are compact, focusing on conditions related to doubling properties and uniform continuity.

Contribution

It provides equivalent conditions for the compactness of averaging operators on non-reflexive Lebesgue spaces in metric measure spaces with specific properties.

Findings

Averaging operators are compact under certain doubling and continuity conditions.

Equivalent criteria for compactness are established for $L^1$ and $L^ty$ spaces.

Results extend understanding of operator compactness in non-reflexive Lebesgue spaces.

Abstract

Let $X$ be a Borel and Borel-regular metric measure space whose closed balls are of positive and finite measure. In this paper, we shall give equivalent conditions for averaging operators on non-reflexive Lebesgue spaces $L^1(X)$ and $L^\infty(X)$ on X to be compact, where X has some doubling property and satisfies certain uniform continuity between metric and measure.