Strictly positive optimal Hardy weights on manifolds and graphs via Green potential

Abstract

Consider a subcritical functional $Q$ associated with a $p$-Schr\"odinger operator defined on a domain $\Omega$ in noncompact Riemannian manifold. Employing the supersolution construction, we obtain a family of strictly positive critical Hardy weights for $Q$. The construction is based on the Green potential, with a strictly positive charge, which is assumed to vanish at infinity. An interesting new feature of this family, depending on a parameter $0\leq \varepsilon\leq 1$, is the transition from positive-criticality to null-criticality at $\varepsilon=0$. Moreover, we show that the best Hardy constant for this family is given by $c=[(p-1)/p]^{p-1}$, and that for $\varepsilon=0$ the corresponding Hardy weight is optimal. We furthermore, prove an analogous result on graphs.