Capacitary domination and Fefferman--Phong inequalities for measure potentials on $2$-PI spaces

Abstract

Let $(X,d,\omega )$ be an unbounded complete doubling metric measure space supporting a weak $(1,2)$-Poincaré inequality, and let $\pi$ be a positive Radon measure. For a ball $B=B(x,R)$, suppose that for some $\delta >0$ and $C_{\mathrm{gr}}>0$, \[ π(B(y,r))\le C_{\mathrm{gr}} \left(\frac rR\right)^δ\frac{ω(B(y,r))}{r^2}, \qquad y\in B,\quad 0<r\le R. \] Let ${\mathrm{cap}}_{2,\omega }(K;\Omega )$ denote the relative variational $2$-capacity of $K$ in an open set $\Omega$. We prove that, for every fixed ${\Lambda }_0>2$, there is a constant $C>0$ such that \[ π(K)\le C\operatorname{cap}_{2,ω}(K;Λ_0B), \qquad K\subset B\ \text{compact}, \] with the same outer ball for all compact $K$. No doubling or lower-Ahlfors regularity is imposed on $\pi$. This fixed-domain capacitary domination yields representative-independent $L^2(d\pi )$ traces and, under a lower normalization $\pi (B)\ge c_0\omega (B)/R^2$ with $c_0>0$, the mixed $d\omega d\pi$ Poincaré estimate. Summation over bounded-overlap normalized covers gives two-sided global Fefferman--Phong inequalities and identifies the corresponding homogeneous energy spaces. In ${\mathbb{R}}^d$ with a Muckenhoupt $A_2$ weight $w$, scale decay and controlled enlargement of $\pi$ generate the required critical-radius cover for every sufficiently large fixed threshold $A$; no additional lower reverse-doubling hypothesis on $w$ is needed. The resulting energy comparison gives equality of the measure-potential and critical-multiplier form domains, a smooth form core, a nonnegative self-adjoint realization, resolvent estimates, and local critical-multiplier bounds. The framework also covers reverse-Hölder function potentials, Carnot groups, and lower-dimensional singular measures, and provides a representative-aware capacity derivation of an earlier generalized Schrödinger energy estimate.