Trace Hardy--Sobolev--Mazy'a inequalities for the half fractional Laplacian

TL;DR

This paper establishes optimal trace Hardy-Sobolev-Maz'ya inequalities for the spectral half Laplacian on weakly mean convex domains, introducing a new weighted Hardy estimate and addressing a previously unresolved critical case.

Contribution

It introduces a novel weighted Hardy estimate and extends Hardy-Sobolev-Maz'ya inequalities to the spectral half Laplacian, covering an open critical case.

Findings

Optimal trace Hardy-Sobolev-Maz'ya inequalities established

New weighted Hardy estimate developed

Addresses critical case previously left open

Abstract

In this work we establish trace Hardy-Sobolev-Maz'ya inequalities with best Hardy constants, for weakly mean convex domains. We accomplish this by obtaining a new weighted Hardy type estimate which is of independent inerest. We then produce Hardy-Sobolev-Maz'ya inequalities for the spectral half Laplacian. This covers a critical case left open in \cite{FMT1}.