How Arzel\`a and Ascoli would have proved Pego theorem for $L^1(G)$ (if they lived in the $21^{st}$ century)?

TL;DR

This paper explores how classical Arzelà-Ascoli theorems could be adapted to prove Pego's theorem for $L^1(G)$, bridging historical compactness characterizations with harmonic analysis techniques.

Contribution

It provides a novel connection between Arzelà-Ascoli and Pego theorems, extending classical compactness results to the setting of $L^1(G)$ on locally compact abelian groups.

Findings

Established a theoretical link between Arzelà-Ascoli and Pego theorems.

Extended classical compactness characterizations to harmonic analysis context.

Proposed a framework for understanding compact families in $L^1(G)$.

Abstract

In the paper we make an effort to answer the question ``What if Arzel\`a and Ascoli lived long enough to see Pego theorem?''. Giulio Ascoli and Cesare Arzel\`a died in 1896 and 1912, respectively, so they could not appreciate the characterization of compact families in $L^2(\mathbb{R}^N)$ provided by Robert L. Pego in 1985. Unlike the Italian mathematicians, Pego employed various tools from harmonic analysis in his work (for instance the Fourier transform or the Hausdorff-Young inequality). Our article is meant to serve as a bridge between Arzel\`a-Ascoli theorem and Pego theorem (for $L^1(G)$ rather than $L^2(G)$, $G$ being a locally compact abelian group). In a sense, the former is the ``raison d'\^etre'' of the latter, as we shall painstakingly demonstrate.