On the Attainability of the Best Constant in Fractional Hardy-Sobolev Inequalities Involving the Spectral Dirichlet Laplacian

Abstract

We prove the attainability of the best constant in the fractional Hardy-Sobolev inequality with a boundary singularity for the spectral Dirichlet Laplacian. The main assumption is the average concavity of the boundary at the origin. A similar result has been proved earlier for the conventional Hardy-Sobolev inequality.