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

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 $\Omega$.