Optimal hardy weight for second-order elliptic operator: An answer to a problem of Agmon

Abstract

For a general subcritical second-order elliptic operator P in a domain Ω⊂Rn (or noncompact manifold), we construct Hardy-weight W which is optimal in the following sense. The operator P−λW is subcritical in Ω for all λ<1, null-critical in Ω for λ=1, and supercritical near any neighborhood of infinity in Ω for any λ>1. Moreover, if P is symmetric and W>0, then the spectrum and the essential spectrum of W−1P are equal to [1,∞), 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 Ω.