Optimal Hardy Weight for Second-Order Elliptic Operator: An Answer to a Problem of Agmon

TL;DR

This paper constructs an optimal Hardy weight for second-order elliptic operators, characterizing subcriticality, null-criticality, and supercriticality, with implications for spectral theory and Agmon metrics.

Contribution

It provides an explicit Hardy weight for general subcritical elliptic operators, extending the theory to nonsymmetric cases and linking it to spectral properties and Agmon metrics.

Findings

The Hardy weight is optimal and explicitly constructed.

Spectral and essential spectrum of the weighted operator are identified as [1,∞).

The method applies to both symmetric and nonsymmetric operators.

Abstract

For a general subcritical second-order elliptic operator $P$ in a domain $\Omega \subset \mathbb{R}^n$ (or noncompact manifold), we construct Hardy-weight $W$ which is optimal in the following sense. The operator $P - \lambda W$ is subcritical in $\Omega$ for all $\lambda < 1$, null-critical in $\Omega$ for $\lambda = 1$, and supercritical near any neighborhood of infinity in $\Omega$ for any $\lambda > 1$. Moreover, if $P$ is symmetric and $W>0$, then the spectrum and the essential spectrum of $W^{-1}P$ are equal to $[1,\infty)$, and the corresponding Agmon metric is complete. Our method is based on the theory of positive solutions and applies to both symmetric and nonsymmetric operators. The constructed Hardy-weight is given by an explicit simple formula involving two distinct positive solutions of the equation $Pu=0$, the existence of which depends on the subcriticality of $P$ in $\Omega$.