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

Abstract

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 (0, 1) of the fractional Laplacians. In particular, we give a complete answer in the L (2) 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)