Addendum to "The Kolmogorov--Riesz compactness theorem" (Expo. Math. 28:385--394 (2010))

TL;DR

This paper refines the criteria for total boundedness of subsets in Sobolev spaces when embedded into Lebesgue spaces, extending previous results with improved conditions.

Contribution

It provides an improved version of a known theorem on compactness in Sobolev spaces, building on prior work in Morrey–Sobolev spaces.

Findings

Enhanced conditions for total boundedness in Sobolev embeddings

Extension of previous compactness criteria

Application to Morrey–Sobolev spaces

Abstract

We show how to improve on Theorem 10 in [arXiv:0906.4883], describing when subsets in $W^{1,p}(\mathbb{R}^n)$ are totally bounded subsets of $L^q(\mathbb{R}^n)$ for $p<n$ and $p\le q<p^*$. This improvement was first shown by Dosso, Fofana, and Sanogo (2013) in the context of Morrey--Sobolov spaces.