Generalized Harmonic Measures: Synchronized Approximation, Representation and Quantitative Measure Recovery
Abstract
We study synchronized boundary and kernel approximation for harmonic measure systems satisfying nested mean-value identities. Under boundary concentration along a common subsequence, joint limits yield representation on a common refinement, and gluing gives uniqueness on the minimal boundary. Neighborhood nonvanishing gives full-sequence synchronized selection. For a compact set whose removal from a bounded C1,1 domain in dimension at least three leaves a connected domain, zero Newtonian capacity is equivalent to concentration of one normalized singular kernel, constructed from harmonic measures, on a subsequence of one relatively compact exhaustion. For such zero-capacity sets, capacitary estimates give quantitative recovery of finite signed source measures on every prescribed relatively compact exhaustion. On rectifiable curve networks in three dimensions, the logarithmic error bound is sharp for this recovery formula in the class of finite measures. Boundary splitting and finite weighted metric graphs illustrate the gluing conclusions.
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BibTeX
@misc{liu2026generalized,
title = {{Generalized Harmonic Measures: Synchronized Approximation, Representation and Quantitative Measure Recovery}},
author = {Duchao Liu and Yunjie Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.31279},
eprint = {2609.31279},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31279},
}
RIS
TY - UNPB
TI - Generalized Harmonic Measures: Synchronized Approximation, Representation and Quantitative Measure Recovery
AU - Duchao Liu
AU - Yunjie Wang
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2609.31279
UR - https://arxiv.org/abs/2609.31279
AB - We study synchronized boundary and kernel approximation for harmonic measure systems satisfying nested mean-value identities. Under boundary concentration along a common subsequence, joint limits yield representation on a common refinement, and gluing gives uniqueness on the minimal boundary. Neighborhood nonvanishing gives full-sequence synchronized selection. For a compact set whose removal from a bounded C^1,1 domain in dimension at least three leaves a connected domain, zero Newtonian capacity is equivalent to concentration of one normalized singular kernel, constructed from harmonic measures, on a subsequence of one relatively compact exhaustion. For such zero-capacity sets, capacitary estimates give quantitative recovery of finite signed source measures on every prescribed relatively compact exhaustion. On rectifiable curve networks in three dimensions, the logarithmic error bound is sharp for this recovery formula in the class of finite measures. Boundary splitting and finite weighted metric graphs illustrate the gluing conclusions.
ER -