On the best constant of Hardy-Sobolev Inequalities

TL;DR

This paper determines the exact optimal constant for a Hardy-Sobolev inequality involving distance to the origin, revealing cases where it matches the best Sobolev constant, and connects to a limiting Caffarelli-Kohn-Nirenberg inequality.

Contribution

It provides the sharp constant for the Hardy-Sobolev inequality related to the origin distance, including special cases where it equals the best Sobolev constant.

Findings

Sharp constant for Hardy-Sobolev inequality obtained.

In three dimensions, the constant coincides with the best Sobolev constant.

Links to a limiting Caffarelli-Kohn-Nirenberg inequality.

Abstract

We obtain the sharp constant for the Hardy-Sobolev inequality involving the distance to the origin. This inequality is equivalent to a limiting Caffarelli-Kohn-Nirenberg inequality. In three dimensions, in certain cases the sharp constant coincides with the best Sobolev constant.