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 $s\in (0,1)$ of the fractional Laplacians. We answer in particular an open problem raised by Frank and Seiringer \cite{FS}.

42 pages

References in corpus (1)

• $L^1$ Hardy inequalities with weights

Cited by in corpus (7)

• Improved L${}^p$-Poincaré inequalities on the hyperbolic space
• A Strong Maximum Principle for the fractional Laplace equation with mixed boundary condition
• A fractional Laplacian problem with mixed singular nonlinearities and nonregular data
• A note on the existence of a unique positive entropy solution to a fractional Laplacian with singular nonlinearities
• Nonlocal Lazer-McKenna type problem perturbed by the Hardy's potential and its parabolic equivalence
• A geometrical approach to the sharp Hardy inequality in Sobolev-Slobodecki\uı spaces
• Trace Hardy inequality for the Euclidean space with a cut and its applications