On the compactness of the non-radial Sobolev space

Abstract

In this paper, we give the affirmative answer of the question in [17], which is a compactness result of the non-radial Sobolev space. As an application, we show the existence of an extremal function of the critical Hardy inequality under spherical average zero. In Appendix, we give an alternative proof of Hardy type inequalities under spherical average zero.