2026/07/19 by Tan Duc Do
#math.FA
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 π yields π(K)\lesssimcap2,ω(K;Λ0B) uniformly for compact K⊂ B. No doubling or lower-dimensional regularity is imposed on π. This capacitary domination gives a representative-independent mean-zero trace inequality. On normalized balls, it is equivalent, modulo the ambient Poincaré energy, to the mixed dω dπ 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 A2 weights with generalized Schrödinger measure potentials, reverse-Hölder function potentials, Carnot groups, and lower-dimensional singular measures. We compare the local trace conclusion with existing two-weighted Poincaré 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 A2 generalized Schrödinger argument and gives a fixed-dilate finite-scale A2 extension, for the naturally augmented measure, of a later A1 theory.