Different degrees of non-compactness for optimal Sobolev embeddings

TL;DR

This paper quantitatively analyzes the non-compactness of optimal Sobolev embeddings into Lebesgue and rearrangement-invariant spaces, providing sharp estimates and revealing differing degrees of strict singularity.

Contribution

It offers the first quantitative study of the non-compactness structure of these embeddings, including sharp Bernstein number estimates and distinctions between Lebesgue and rearrangement-invariant cases.

Findings

Optimal Sobolev embedding into Lebesgue spaces is finitely strictly singular.

Embedding into all rearrangement-invariant spaces is not strictly singular.

Sharp two-sided Bernstein number estimates are established.

Abstract

The structure of non-compactness of optimal Sobolev embeddings of $m$-th order into the class of Lebesgue spaces and into that of all rearrangement-invariant function spaces is quantitatively studied. Sharp two-sided estimates of Bernstein numbers of such embeddings are obtained. It is shown that, whereas the optimal Sobolev embedding within the class of Lebesgue spaces is finitely strictly singular, the optimal Sobolev embedding in the class of all rearrangement-invariant function spaces is not even strictly singular.