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

Abstract

In the paper we make an effort to answer the question ``What if Arzelà and Ascoli lived long enough to see Pego theorem?''. Giulio Ascoli and Cesare Arzelà 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 Arzelà-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'être'' of the latter, as we shall painstakingly demonstrate.