Optimal Hardy-type inequalities for elliptic operators

Abstract

For a general second order elliptic operator P in a domain Ω, we construct a Hardy weight W in the punctured domain Ω⋆:=Ω∖{0} such that P−λW is subcritical in Ω⋆ for λ<1, null-critical in Ω⋆ for λ=1, and supercritical near infinity and near 0 for λ>1. 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 formula involving the Green function of P and its gradient.