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

TL;DR

This paper establishes optimal trace Hardy and Hardy-Sobolev-Maz'ya inequalities for domains with geometric conditions, and applies these to derive fractional Laplacian inequalities, solving an open problem for the half-space case.

Contribution

It introduces sharp trace Hardy and Hardy-Sobolev-Maz'ya inequalities for various domains and fractional Laplacians, including the full range of s in (0,1).

Findings

Established best Hardy constants for trace inequalities.

Derived fractional Hardy-Sobolev-Maz'ya inequalities for fractional Laplacians.

Solved an open problem for the half-space case.

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}.