复 Monge-Ampere 测度的预定极限与聚点集
Prescribed Limits and Cluster Sets of Complex Monge-Ampere Measures
- Chern Institute of Mathematics and LPMC, Nankai University（南开大学陈省身数学研究所和LPMC）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本研究针对 Bedford 问题，刻画了有界多重次调和序列的 Monge-Ampère 测度在模糊拓扑下的聚点集，并构造逼近序列实现预定测度极限，推广了经典结果。
AI中文摘要：
受 Bedford 问题的启发，我们研究了在模糊拓扑下，与有界多重次调和序列 $u_j\to\varphi$（局部在 $L^1$ 中）相关的 Monge–Ampère 测度可能的聚点集，而不假设公共界。在有界域 $\Omega\subset\mathbb C^n$（$n\geq2$）上，对于有界极大多重次调和函数 $\varphi$，所有模糊子序列测度极限的可能集合恰好是包括空集在内的非负 Radon 测度的模糊闭集。当 $\Omega$ 是超凸、伪凸且 Runge 域，或从有界超凸域中移除相对闭的多次调和集所得的域时，该结论对所有有界多重次调和 $\varphi$ 成立。在这些假设下，每个非负 Radon 测度 $\nu$ 也可由这样的序列实现，使得 $(dd^cu_j)^n\to\nu$ 模糊收敛。主要解析结果在每个有界域上成立：对于任意有界多重次调和 $\varphi$ 和非负 Radon 测度 $\mu$，我们构造 $u_j\leq\varphi$ 且对每个 $1\leq p<\infty$ 在 $L^p(\Omega)$ 中收敛到 $\varphi$，并且 $(dd^cu_j)^n\to(dd^c\varphi)^n+\mu$ 模糊收敛。当 $\varphi$ 连续时，逼近函数可选为光滑的。在任意伪凸域上，预定测度极限也可用局部有界逼近函数实现。在附加局部化条件下，当极限函数有界时，逼近函数可选为全局有界的。
英文摘要：
Motivated by Bedford's question, we study the possible cluster sets, in the vague topology, of Monge--Ampère measures associated with bounded plurisubharmonic sequences $u_j\toφ$ locally in $L^1$, without assuming a common bound. On a bounded domain $Ω\subset\mathbb C^n$ ($n\geq2$), and for a bounded maximal plurisubharmonic $φ$, the possible sets of all vague subsequential measure limits are precisely the vaguely closed sets of nonnegative Radon measures, including the empty set. This conclusion holds for every bounded plurisubharmonic $φ$ when $Ω$ is hyperconvex, pseudoconvex and Runge, or a domain obtained by removing a relatively closed pluripolar set from a bounded hyperconvex domain. Under these hypotheses, every nonnegative Radon measure $ν$ can also be realized by such a sequence with $(dd^cu_j)^n\toν$ vaguely. The main analytic result holds on every bounded domain: for any bounded plurisubharmonic $φ$ and nonnegative Radon measure $μ$, we construct $u_j\leqφ$ converging to $φ$ in $L^p(Ω)$ for every $1\leq p<\infty$, with $(dd^cu_j)^n\to(dd^cφ)^n+μ$ vaguely. The approximants can be chosen smooth when $φ$ is continuous. On arbitrary pseudoconvex domains, prescribed measure limits can also be realized using locally bounded approximants. Under an additional localization condition, the approximants can be chosen globally bounded when the limit function is bounded.
