Hardy-Sobolev-Maz'ya inequalities for arbitrary domains

TL;DR

This paper establishes Hardy-Sobolev-Maz'ya inequalities for any domain in rom which new spectral bounds for Schrf6dinger operators are derived, confirming a conjecture for convex domains.

Contribution

It proves a Hardy-Sobolev-Maz'ya inequality for arbitrary domains, including convex ones, with a universal constant, and applies this to eigenvalue estimates of Schrf6dinger operators.

Findings

Hardy-Sobolev-Maz'ya inequality holds for all domains in or Ngeq 3

Confirmed a conjecture for convex domains

Derived Hardy-Lieb-Thirring inequalities for eigenvalues

Abstract

We prove a Hardy-Sobolev-Maz'ya inequality for arbitrary domains \Omega\subset\R^N with a constant depending only on the dimension N\geq 3. In particular, for convex domains this settles a conjecture by Filippas, Maz'ya and Tertikas. As an application we derive Hardy-Lieb-Thirring inequalities for eigenvalues of Schr\"odinger operators on domains.