Characterizing compact sets in $L^p$-spaces and its application

TL;DR

This paper generalizes the Kolmogorov-Riesz theorem to characterize compact sets in $L^p$-spaces on metric measure spaces and explores the topology of Lipschitz maps with bounded supports.

Contribution

It introduces a new criterion for compactness in $L^p$-spaces and applies it to analyze the topological structure of Lipschitz maps.

Findings

Established a generalized compactness criterion in $L^p$-spaces.

Analyzed the topological type of Lipschitz maps with bounded supports.

Extended classical theorems to more general metric measure spaces.

Abstract

In this paper, we give a characterization of compact sets in $L^p$-spaces on metric measure spaces, which is a generalization of the Kolmogorov-Riesz theorem. Using the criterion, we investigate the topological type of the space consisting of lipschitz maps with bounded supports.