Decoupling and Tail Laws for Small-Order Metric-Valued Nonlocal Energies: A Structural View of the Maz'ya--Shaposhnikova Formula
Abstract
We develop a measure-theoretic framework for small-order limits of nonlocal energies with metric-valued maps. For globally Lp maps, we consider interaction measures whose marginals have uniformly bounded L∞ densities aα and bα. If these densities converge weakly-* to a∞ and b∞, the normalized energies converge to the Lp energy weighted by a∞+b∞ if and only if the interaction measures escape every bounded rectangle. For maps that are only locally Lp, we introduce target laws describing the distribution of the values sampled at infinity. Convergence of their p-distance profiles yields the limiting interaction energy for every 1≤ p<∞ and arbitrary Polish targets, with integrated p-Wasserstein convergence as a sufficient criterion. The arguments include the endpoint p=1 and require no linear structure on the target space. Applications include directional and anisotropic Maz'ya--Shaposhnikova formulas, a Bernoulli-law interpretation of fractional perimeters, and an Abelian principle for heat-semigroup energies.
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BibTeX
@misc{pinamonti2026decoupling,
title = {{Decoupling and Tail Laws for Small-Order Metric-Valued Nonlocal Energies: A Structural View of the Maz'ya--Shaposhnikova Formula}},
author = {Andrea Pinamonti},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01958},
eprint = {2610.01958},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01958},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Decoupling and Tail Laws for Small-Order Metric-Valued Nonlocal Energies: A Structural View of the Maz'ya--Shaposhnikova Formula
AU - Andrea Pinamonti
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01958
UR - https://arxiv.org/abs/2610.01958
AB - We develop a measure-theoretic framework for small-order limits of nonlocal energies with metric-valued maps. For globally L^p maps, we consider interaction measures whose marginals have uniformly bounded L^∞ densities a_α and b_α. If these densities converge weakly-* to a_∞ and b_∞, the normalized energies converge to the L^p energy weighted by a_∞+b_∞ if and only if the interaction measures escape every bounded rectangle. For maps that are only locally L^p, we introduce target laws describing the distribution of the values sampled at infinity. Convergence of their p-distance profiles yields the limiting interaction energy for every 1≤ p<∞ and arbitrary Polish targets, with integrated p-Wasserstein convergence as a sufficient criterion. The arguments include the endpoint p=1 and require no linear structure on the target space. Applications include directional and anisotropic Maz'ya--Shaposhnikova formulas, a Bernoulli-law interpretation of fractional perimeters, and an Abelian principle for heat-semigroup energies.
ER -