Quality of non-compactness for Sobolev Embedding with one point non-compactness

TL;DR

This paper analyzes the non-compactness of Sobolev embeddings with a single non-compact point, establishing sharp conditions, and providing bounds for various measures of non-compactness, revealing infinite-dimensional invertible subspaces.

Contribution

It introduces sharp conditions for non-compactness in Sobolev embeddings with one non-compact point and explores the structure of the resulting subspaces.

Findings

Identifies sharp conditions distinguishing compactness from non-compactness.

Shows existence of infinite-dimensional subspaces where the embedding is invertible.

Provides lower bounds for Bernstein, entropy numbers, and non-compactness measure.

Abstract

This paper investigates instances of Sobolev embeddings characterized by local compactness at every point within their domain, except for a single point. We obtain the sharp conditions that distinguish compactness from non-compactness and observe that in the context of Sobolev embeddings, non-compactness occurring at only one point within the domain could give rise to an infinite-dimensional subspace where the embedding is invertible (i.e., not strictly singular). Furthermore, we establish lower bounds for the Bernstein numbers, entropy numbers, and the measure of non-compactness.