Matrix weighted Kolmogorov-Riesz's compactness theorem

TL;DR

This paper extends the Kolmogorov-Riesz compactness theorem to weighted Lebesgue spaces with matrix weights, providing new characterizations of totally bounded sets when weights are in the $A_p$ class.

Contribution

It introduces several versions of the theorem in matrix-weighted spaces and characterizes totally bounded subsets for $A_p$ class weights.

Findings

Characterization of totally bounded subsets in $L^p(W)$ for matrix weights in $A_p$.

Extension of the Kolmogorov-Riesz theorem to matrix-weighted Lebesgue spaces.

New versions of the compactness theorem in weighted spaces.

Abstract

In this paper, several versions of the Kolmogorov-Riesz compactness theorem in weighted Lebesgue spaces with matrix weights are obtained. In particular, when the matrix weight $W$ is in the known $A_p$ class, a characterization of totally bounded subsets in $L^p(W)$ with $p\in(1, \infty)$ is established.