Compactness of averaging operators on 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.