  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.
