A spectral result for Hardy inequalities

Abstract

Let P be a linear, second order, elliptic operator satisfying a Hardy inequality with potential W (i.e. $P-W\ge 0$) 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.

To appear in J. Math Pures Appl. This version: extensive changes in the exposition, among which: a new introduction, and a new section about spectrum and Agmon metrics. New results and examples have also been added