Strict s-numbers of non-compact Sobolev embeddings into continuous functions

TL;DR

This paper investigates the behavior of various strict s-numbers in non-compact Sobolev embeddings into continuous functions, revealing their asymptotic properties and finite strict singularity in limiting cases.

Contribution

It extends existing results on s-numbers of Sobolev embeddings to the non-compact limiting case, providing sharp estimates and revealing new behaviors.

Findings

Exact s-number values in 1D cases.

Sharp asymptotic estimates in higher dimensions.

Limiting embeddings are finitely strictly singular.

Abstract

For limiting non-compact Sobolev embeddings into continuous functions we study behavior of Approximation, Gelfand, Kolmogorov, Bernstein and Isomorphism s-numbers. In the one dimensional case the exact values of the above-mentioned strict s-numbers were obtained and in the higher dimensions sharp estimates for asymptotic behavior of strict s-numbers were established. As all known results for s-numbers of Sobolev type embeddings are studied mainly under the compactness assumption then our work is an extension of existing results and reveal an interesting behavior of s-numbers in the limiting case when some of them (Approximation, Gelfand and Kolmogorov) have positive lower bound and others (Bernstein and Isomorphism) are decreasing to zero. From our results also follows that such limiting non-compact Sobolev embeddings are finitely strictly singular maps.