Natural proof of the characterization of relatively compact families in $L^p-$spaces on locally compact groups

TL;DR

This paper provides a natural and elegant proof for characterizing relatively compact families in $L^p$-spaces on locally compact groups, using convolution to connect properties in $L^p$ and $C_0$ spaces.

Contribution

It introduces a novel proof approach that leverages convolution with compactly supported functions to relate relative compactness in $L^p$ and $C_0$ spaces.

Findings

Convolution preserves key properties of families in $L^p$-spaces.

Relative compactness in $C_0$ implies relative compactness in $L^p$.

The proof simplifies existing characterizations of relative compactness.

Abstract

In the paper we look for an elegant proof of the characterization of relatively compact families in $L^p-$spaces. At first glance, the suggested approach may seem convoluted and lengthy, but we spare no effort to argue that our proof is in fact more natural than the ones existent in the literature. The key idea is that the three properties which characterize relative compactness in $L^p-$spaces ($L^p-$boundedness, $L^p-$equicontinuity and $L^p-$equivanishing) are "preserved" (or rather "inherited") when the family $\Ffamily\subset L^p$ is convolved with a continuous and compactly supported function. This new family turns out to be relatively compact in $C_0-$space and it remains to be demonstrated that relative compactness in $C_0-$space implies relative compactness of the original family $\Ffamily.$