Mixed Poincar\'e and Fefferman--Phong inequalities for measure potentials on $2$-PI spaces
We prove a fixed-outer-domain content--capacity estimate for every positive codimensional gain on an unbounded complete $2$-PI space. As its principal measure-theoretic consequence, a one-sided ball-growth condition for an arbitrary positive Radon measure $\pi$ yields \[ \pi(K)\lesssim\operatorname{cap}_{2,\omega}(K;\Lambda_0B) \] uniformly for compact $K\subset B$. No doubling or lower-dimensional regularity is imposed on $\pi$. This capacitary domination gives a representative-independent mean-zero trace inequality. On normalized balls, it is equivalent, modulo the ambient Poincar\'e energy, to the mixed $d\omega\,d\pi$ oscillation required in Fefferman--Phong reductions. A standard bounded-overlap localization then yields two-sided global Fefferman--Phong inequalities, cover-independent energy norms, and a concrete realization of the associated homogeneous energy completion. The abstract results are verified for Euclidean $A_2$ weights with generalized Schr\"odinger measure potentials, reverse-H\"older function potentials, Carnot groups, and lower-dimensional singular measures. We compare the local trace conclusion with existing two-weighted Poincar\'e and Sobolev embedding theorems: those routes apply under additional doubling or dimension assumptions on the target measure, whereas our capacity conclusion also supplies absolute continuity with respect to variational capacity and a fixed outer domain. The Euclidean application yields form-domain equivalence, smooth form cores, self-adjoint realization, resolvent energy estimates, and local critical-multiplier bounds. Finally, the method supplies the mixed-measure step missing from a previously published $A_2$ generalized Schr\"odinger argument and gives a fixed-dilate finite-scale $A_2$ extension, for the naturally augmented measure, of a later $A_1$ theory.