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arXiv 2207.13610math.CV
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Monge-Ampère测度的一个等式

An equality of Monge-Ampère measures

Mohamed El Kadiri

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8 pages
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摘要：

设$u$和$v$是整环$Ω\subset {\bf C}^n$上Monge-Ampère算子定义域中的两个多重次调和函数.证明了若$u=v$在多细开集$U\subset Ω$上是Borel可测的，则$(dd^cu)^n|_U=(dd^cv)^n|_U$ .这个结果在$u$和$v$是局部有界的情况下由Bedford和Taylor证明，在$u$和$v$是有限的情况下由El Kadiri和Wiegerinck证明，在$U$是$U=\bigcup_{j=1}^m\{φ_j>ψ_j\}$形式的情况下由Hai和Hiep证明，其中$φ_j$，$ψ_j$，$j=1,...,m$是$Ω$上的多重次调和函数。

英文摘要：

Let $u$ and $v$ be two plurisubharmonic functions in the domain of definition of the Monge-Ampère operator on a domain $Ω\subset {\bf C}^n$. We prove that if $u=v$ on a plurifinely open set $U\subset Ω$ that is Borel measurable, then $(dd^cu)^n|_U=(dd^cv)^n|_U$. This result was proved by Bedford and Taylor in the case where $u$ and $v$ are locally bounded, and by El Kadiri and Wiegerinck when $u$ and $v$ are finite, and by Hai and Hiep when $U$ is of the form $U=\bigcup_{j=1}^m\{φ_j>ψ_j\}$, where $φ_j$, $ψ_j$, $j=1,...,m$, are plurisubharmonic functions on $Ω$.

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