发表机构
Università degli Studi di Trento（特伦托大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出测度论框架，研究度量值非局部能量的小阶极限，给出收敛的充要条件，并应用于Maz'ya--Shaposhnikova公式、分数周长及热半群能量。
AI 中文摘要
我们为度量值映射的非局部能量的小阶极限建立了一个测度论框架。对于全局$L^p$映射，我们考虑交互测度，其边际具有一致有界的$L^\infty$密度$a_\alpha$和$b_\alpha$。如果这些密度弱-*收敛到$a_\infty$和$b_\infty$，则归一化能量收敛到由$a_\infty+b_\infty$加权的$L^p$能量，当且仅当交互测度逃离每个有界矩形。对于仅局部$L^p$的映射，我们引入目标定律来描述在无穷远处采样的值的分布。其$p$-距离轮廓的收敛性产生了对于每个$1\leq p<\infty$和任意Polish目标的极限交互能量，其中积分$p$-Wasserstein收敛作为充分条件。论证包括端点$p=1$，且不需要目标空间上的线性结构。应用包括方向性和各向异性的Maz'ya--Shaposhnikova公式、分数周长的一个Bernoulli定律解释，以及热半群能量的一个Abel原理。
英文摘要
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 $L^\infty$ densities $a_α$ and $b_α$. If these densities converge weakly-* to $a_\infty$ and $b_\infty$, the normalized energies converge to the $L^p$ energy weighted by $a_\infty+b_\infty$ 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 $1\leq p<\infty$ 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.
