On the compactness of the (non)radial Sobolev spaces

TL;DR

This paper establishes a compactness result for non-radial Sobolev spaces, demonstrates the existence of extremal functions for the critical Hardy inequality with spherical average zero, and improves existing compactness results for radial Sobolev spaces.

Contribution

It provides a positive answer to a previously open question on non-radial Sobolev space compactness and enhances the understanding of Hardy inequalities under spherical average zero.

Findings

Confirmed compactness of non-radial Sobolev spaces

Proved existence of extremal functions for Hardy inequality

Improved compactness results for radial Sobolev spaces

Abstract

In this note, we give the affirmative answer of the question in [18], which is a compactness result of the non-radial Sobolev spaces. As an application, we show the existence of an extremal function of the critical Hardy inequality under spherical average zero. Next, we give an improvement of the compactness results of the radial Sobolev spaces in [8]. In Appendix, we give an alternative proof of Hardy type inequalities under spherical average zero.