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A comparison principle for the complex Monge-Ampère operator in Cegrell’s classes and applications. (English) Zbl 1179.32014

Let \(\Omega\) be a bounded hyperconvex domain in \({\mathbb C}^n\). A plurisubharmonic function on \(\Omega\) is said to belong to the Cegrell class \({\mathcal F}\) if it is the limit of a decreasing sequence of bounded plurisubharmonic functions with uniformly bounded Monge-Ampère masses on \(\Omega\) and zero boundary values on \(\partial\Omega\), and it belongs to the Cegrell class \(\mathcal E\) if it is locally in \(\mathcal F\).
The authors establish useful integral bounds on functions from the Cegrell classes, in the spirit of the comparison principle from Y. Xing [Proc. Am. Math. Soc. 124, No. 2, 457–467 (1996; Zbl 0849.31010)]. For example, if \(u,v\in{\mathcal F}\) are such that \(u\leq v\), then
\[ \int_\Omega(v-u)^ndd^cw_1\wedge\dots\wedge dd^cw_n\leq n! \int_\Omega|w_1|[(dd^cu)^n-(dd^cv)^n] \]
for all plurisubharmonic functions \(w_j\), \(-1\leq w_j\leq 0\) (Prop. 3.4). As a consequence, if \(u,u_j\in{\mathcal F}\), \(u_j\leq u\), and the Monge-Ampère masses of \(u_j\) on \(\Omega\) are uniformly bounded, then \((dd^cu_j)^n\to(dd^cu)^n\) by variation if and only if \(u_j\to u\) in \(C_n\)-capacity on compact subsets of \(\Omega\) (Thm. 3.5).
Another central result is Thm. 4.1: If \(u,u_1,\dots,u_{n-1}\in{\mathcal E}\), \(v\in PSH^-(\Omega)\), \(T=dd^cu_1\wedge\dots\wedge dd^cu_{n-1}\), then \(dd^c\max(u,v)\wedge T|_{\{u>v\}}=dd^cu\wedge T|_{\{u>v\}}.\) It implies more Xing-type comparison theorems for functions from the classes \(\mathcal F\) and \(\mathcal E\) (Thm. 4.7, 4.9).

MSC:

32W20 Complex Monge-Ampère operators
32U15 General pluripotential theory

Citations:

Zbl 0849.31010
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Full Text: DOI arXiv

References:

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