Compactness of averaging operators on Banach function spaces

Let X be a Borel metric measure space such that every closed ball is of positive and finite measure. In this paper, we give a sufficient condition and a necessary condition for averaging operators on a Banach function space E(X) on X to be compact. As a corollary, we show that the averaging operators on the Lorentz space \(L^{p,q}(X)\) $L^{p,q}(X)$ of X are compact if and only if X is bounded, in the case where X is a doubling and Borel-regular metric measure space with some continuity between metric and measure.