Improved Hardy-Sobolev Inequality under Moment Constraints

Abstract

This paper is inspired by Aubin's 1979 result, which established that the best constant in the Sobolev inequality on the n-sphere, S^n, can be improved under the condition of vanishing first-order moments. Recent advancements by Hang and Wang (2021) showed that Aubin's improvement can be generalized to arbitrary higher-order moments. We further extend Hang and Wang's results to the Hardy-Sobolev inequality on S^n by deriving an associated concentration-compactness principle and imposing similar moment constraints. Finally, we briefly outline a framework for extending these results to higher-order Sobolev spaces.