A spectral result for Hardy inequalities

TL;DR

This paper establishes spectral conditions for elliptic operators satisfying Hardy inequalities, identifying when the spectrum of a related operator begins at the best Hardy constant, with applications to classical inequalities.

Contribution

It provides new spectral criteria for Hardy inequalities and applies these to classical and geometric Hardy inequalities, linking spectral theory with inequality optimization.

Findings

Spectrum of W^{-1}P is [α, ∞) under certain conditions

Identifies spectral properties for Hardy inequalities on convex domains

Analyzes Hardy inequalities for minimal submanifolds

Abstract

Let P be a linear, second order, elliptic operator satisfying a Hardy inequality with potential W (i.e. $P-W\geq0$) and best constant $\alpha$. We give conditions so that the spectrum of $W^{-1}P$ is $[\alpha,\infty)$. We apply this to several well-known Hardy inequalities: (improved) Hardy inequalities on a bounded convex domain with potential involving the distance to the boundary, and Hardy inequalities for minimal submanifolds of the Euclidean space.