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Maximal subextensions of plurisubharmonic functions. (English. French summary) Zbl 1251.32026

Let \(X\) be a complex manifold of dimension \(n\) and \(D\subset \Omega\subset X\) be domains. A subextension of \(u\in {\mathcal PSH}(D)\) to \(\Omega\) is a function \(v\in {\mathcal PSH}(\Omega)\) such that \(v|_D\leq u\). If a subextension of \(u\) to \(\Omega\) exists, then the maximal subextension \(\tilde u\) of \(u\) to \(\Omega\) is defined as
\[ \tilde u=\sup \big\{v\in {\mathcal PSH}(\Omega)\,;\, v|_D\leq u\big\}. \]
The paper is a continuation of [Math. Z. 250, No. 1, 7–22 (2005; Zbl 1080.32032)] by the same authors and the purpose is to study the Monge-Ampère measure \((dd^c\tilde u)^n\) of maximal subextensions.
First it is assumed that \(X={\mathbb C}^n\), \(D\) and \(\Omega\) are bounded hyperconvex domains, and \(D\) is relatively compact in \(\Omega\). U. Cegrell and A. Zeriahi [C. R. Acad. Sci Paris 336, No. 4, 305–308 (2003; Zbl 1025.31005)] proved that every \(u\in {\mathcal F}(D)\) admits a subextension \(v\) to \(\Omega\), and it follows then that \(v\in {\mathcal F}(\Omega)\). Here \({\mathcal F}(D)\) is the set of all \(\varphi\in {\mathcal PSH}(D)\) such that there exists a sequence \((\varphi_j)\) in \({\mathcal E}_0(D)\) with \(\varphi_j\searrow \varphi\) and \(\sup_j \int_D(dd^c\varphi_j)^n<+\infty\), where \({\mathcal E}_0(D)\) is the class of all \(\psi\in {\mathcal PSH}(D)\cap L^\infty(D)\) with \(\psi \leq 0\) and \(\lim_{z\to p}\psi(z)=0\) for all \(p\in \partial D\) and \(\int_D (dd^c\psi)^n<+\infty\).
The first main result is Theorem 2.1: If \(u\in {\mathcal F}(U)\), then \(\tilde u\in {\mathcal F}(\Omega)\), \((dd^c\tilde u)^n \leq {\mathbf 1}_D (dd^cu)^n\), and \(\int_{\tilde u<u}(dd^c\tilde u)^n=0\), where \({\mathbf 1}_D\) is the characteristic function of the set \(D\).
The authors also look at a Kähler manifold \(X\) with Kähler form \(\omega\) and define \({\mathcal PSH}(D,\omega)\) as the set of all \(\omega\)-plurisubharmonic functions on open subsets \(D\) of \(X\). A domain \(D\) in \(X\) is said to be quasi-hyperconvex if \(D\) admits an exhaustion \(\varrho\in C(D)\cap {\mathcal PSH}(D,\omega)\) taking values in \([-1,0[\). For any \(\varphi \in {\mathcal PSH}(D,\omega)\) define \(\omega_\varphi =dd^c\varphi +\omega\). Then the \(n\)-th wedge power \(\omega_\varphi^n\) is a well defined positive current on \(D\) of bidegree \((n,n)\). The class \({\mathcal F}(D,\omega)\) consists of \(\varphi\in {\mathcal PSH}(D,\omega)\) which are limits of decreasing sequences \((\varphi_j)\) of functions in \({\mathcal P}_0(D,\omega)\) with \(\sup \int_D\omega_{\varphi_j}^n<+\infty\), and \({\mathcal P}_0(D,\omega)\) is the class of \(\psi\in {\mathcal PSH}(D,\omega)\cap L^\infty(D)\), such that \(\psi\leq 0\), \(\lim_{z\to p}\psi(z)=0\) for all \(p\in \partial D\), and \(\int_D \omega_\psi^n<+\infty\). In Lemma 4.2 it is proved that for \(\varphi\in{\mathcal F}(D,\omega)\) \(M_D(\varphi)=\lim_{j\to \infty}\int_D\, \omega_{\varphi_j}^n =\sup_j\, \int_D\, \omega_{\varphi_j}^n\) is independent of the choice of sequence \((\varphi_j)\) in \({\mathcal P}_0(D,\omega)\) converging to \(\varphi\) and it is defined as the Monge-Ampère mass of the function \(\varphi\).
A weight function is an increasing function \(\chi:{\mathbb R}\to {\mathbb R}\) such that \(\chi(t)=t\) for \(t\geq 0\) and \(\chi(-\infty)=-\infty\). For every weight function \(\chi\) the class \({\mathcal E}_\chi(D,\omega)\) consists of all \(\varphi\in {\mathcal PSH}(D,\omega)\) for which there exists a sequence \((\varphi_j)\) in \({\mathcal P}_0(D,\omega)\) such that \(\varphi_j\searrow \varphi\) and \(\sup_j\int_D|\chi(\varphi_j)|\omega_{\varphi_j}^n<+\infty\).
The authors prove the existence of an \(\omega\)-subextension in Theorem 4.6: Let \(D\) be a quasi-hyperconvex domain and assume that \(\int_D \omega^n <\int_X \omega^n\). Let \(\varphi\in {\mathcal F}(D,\omega)\) be such that \(M_D(\varphi)\leq \int_X\omega^n\). Then there exists \(\psi\in {\mathcal PSH}(X,\omega)\) such that \(\psi|_D\leq \varphi\).
The properties of the Monge-Ampère mass of the maximal extension \(\tilde \varphi\) is described in Theorem 4.7: Let \(D\) be a quasi-hyperconvex domain satisfying \(\int_D \omega^n <\int_X \omega^n\) and let \(\varphi\in {\mathcal E}_\chi(D,\omega)\) be such that \(\int_D\omega_\varphi^n\leq \int_X\omega^n\), where \(\chi\) is a convex weight function. Then the maximal subextension \(\tilde \varphi\) of \(\varphi\) from \(D\) to \(X\) exists and has the following properties:
(i) \(\tilde \varphi\in {\mathcal E}_\chi(X,\omega)\) and \(\int_X|\chi\circ \tilde \varphi|\, \omega_{\tilde \varphi}^n \leq \int_D|\chi\circ \varphi|\, \omega_{\varphi}^n\),
(ii) \({\mathbf 1}_D\, \omega_{\tilde\varphi}^n\leq {\mathbf 1}_D\, \omega_{\varphi}^n\) holds in the sense of measures on \(X\),
(iii) the measure \(\omega_{\tilde \varphi}^n\) is carried by the Borel set \(\{\tilde \varphi=\varphi\}\cup \partial D\).
At the end of the paper the Lelong class is viewed as the class \({\mathcal PSH}({\mathbb P}^n,\omega_{FS})\) where \(\omega_{FS}\) is the normalized Fubini-Study metric and results on subextensions in the Lelong class are given.

MSC:

32U05 Plurisubharmonic functions and generalizations
32W20 Complex Monge-Ampère operators
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References:

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