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 Ω⊂RnΩ⊂Rn (or noncompact manifold), we construct Hardy-weight W which is optimal   in the following sense. The operator P−λWP−λW is subcritical in Ω   for all λ<1λ<1, null-critical in Ω   for λ=1λ=1, and supercritical near any neighborhood of infinity in Ω   for any λ>1λ>1. Moreover, if P   is symmetric and W>0W>0, then the spectrum and the essential spectrum of W−1PW−1P are equal to [1,∞)[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=0Pu=0, the existence of which depends on the subcriticality of P in Ω.