Sharp Trace Hardy-Sobolev-Maz'ya Inequalities and the Fractional Laplacian

In this work we establish trace Hardy and trace Hardy-Sobolev-Maz'ya inequalities with best Hardy constants for domains satisfying suitable geometric assumptions such as mean convexity or convexity. We then use them to produce fractional Hardy-Sobolev-Maz'ya inequalities with best Hardy constants for various fractional Laplacians. In the case where the domain is the half space, our results cover the full range of the exponent $${s \in}$$ (0, 1) of the fractional Laplacians. In particular, we give a complete answer in the L² setting to an open problem raised by FRANK and SEIRINGER ("Sharp fractional Hardy inequalities in half-spaces," in Around the research of Vladimir Maz'ya. International Mathematical Series, 2010).