Capacitary domination and Fefferman--Phong inequalities for measure potentials on $2$-PI spaces
Abstract
Let (X,d,ω) be an unbounded complete doubling metric measure space supporting a weak (1,2)-Poincaré inequality, and let π be a positive Radon measure. For a ball B=B(x,R), suppose that for some δ>0 and Cgr>0, π(B(y,r))≤ Cgr (r/R)δω(B(y,r))/r2, y∈ B, 0<r≤ R. Let cap2,ω(K;Ω) denote the relative variational $2$-capacity of K in an open set Ω. We prove that, for every fixed Λ0>2, there is a constant C>0 such that π(K)≤ Ccap2,ω(K;Λ0B), K⊂ B compact, with the same outer ball for all compact K. No doubling or lower-Ahlfors regularity is imposed on π. This fixed-domain capacitary domination yields representative-independent L2(dπ) traces and, under a lower normalization π(B)≥ c0ω(B)/R2 with c0>0, the mixed dω dπ Poincaré estimate. Summation over bounded-overlap normalized covers gives two-sided global Fefferman--Phong inequalities and identifies the corresponding homogeneous energy spaces. In ℝd with a Muckenhoupt A2 weight w, scale decay and controlled enlargement of π 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.
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BibTeX
@misc{do2026capacitary,
title = {{Capacitary domination and Fefferman--Phong inequalities for measure potentials on $2$-PI spaces}},
author = {Tan Duc Do},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.17315},
eprint = {2607.17315},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.17315},
}
RIS
TY - UNPB
TI - Capacitary domination and Fefferman--Phong inequalities for measure potentials on $2$-PI spaces
AU - Tan Duc Do
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2607.17315
UR - https://arxiv.org/abs/2607.17315
AB - Let (X,d,ω) be an unbounded complete doubling metric measure space supporting a weak (1,2)-Poincaré inequality, and let π be a positive Radon measure. For a ball B=B(x,R), suppose that for some δ>0 and C_gr>0, π(B(y,r))≤ C_gr (r/R)^δω(B(y,r))/r^2, y∈ B, 0<r≤ R. Let cap_2,ω(K;Ω) denote the relative variational $2$-capacity of K in an open set Ω. We prove that, for every fixed Λ_0>2, there is a constant C>0 such that π(K)≤ Ccap_2,ω(K;Λ_0B), K⊂ B compact, with the same outer ball for all compact K. No doubling or lower-Ahlfors regularity is imposed on π. This fixed-domain capacitary domination yields representative-independent L^2(dπ) traces and, under a lower normalization π(B)≥ c_0ω(B)/R^2 with c_0>0, the mixed dω dπ Poincaré estimate. Summation over bounded-overlap normalized covers gives two-sided global Fefferman--Phong inequalities and identifies the corresponding homogeneous energy spaces. In ℝ^d with a Muckenhoupt A_2 weight w, scale decay and controlled enlargement of π 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.
ER -