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

Consider a subcritical functional $Q$ associated with a $p$-Schrödinger operator defined on a domain $\mathrm{\Omega }$ 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 $0\le \epsilon \le 1$, is the transition from positive-criticality to null-criticality at $\epsilon =0$. Moreover, we show that the best Hardy constant for this family is given by $c=[(p-1)\mathrm{/}p{]}^{p-1}$, and that for $\epsilon =0$ the corresponding Hardy weight is optimal. We furthermore, prove an analogous result on graphs.