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Weak continuity of the complex \(k\)-Hessian operators with respect to local uniform convergence. (English) Zbl 1296.32016
Let \(\Omega\) be a domain in \(\mathbb C^n\), and let \(k\) be an integer \(1\leq k\leq n\). A function \(u\in \mathcal C^2(\Omega)\) is called \(k\)-plurisubharmonic if
\[ F_m(u)=F_m\left (\left[\frac {\partial ^2u}{\partial z_j\partial \bar {z}_l}\right]_{j,l=1}^{n}\right)\geq 0\quad\text{for}\quad m\in \{1,\dots,k\}, \] where \(F_m(A)\) is defined as the sum of all \(m\times m\) principal minors of an \(n\times n\) Hermitian matrix \(A\).

A function \(u\in \mathcal C (\Omega)\) is called \(k\)-plurisubharmonic (\(u\in \Phi^k(\Omega)\)) if there exists a sequence of continuous \(k\)-plurisubharmonic functions \(u_j\) such that \(u_j\to u\), \(j\to \infty\), locally uniformly in \(\Omega\).

The aim of this paper is to give a new proof of the well known fact that the complex \(k\)-Hessian operators \(F_k\) are continuous with respect to local uniform convergence. The authors prove the following: for any \(u\in \Phi^k(\Omega)\) there exists a Borel measure \(\mu_k[u]\) such that
\[ \mu_k[u](E)=\int_EF_k(u)dV_{2n} \] if \(u\in \mathcal C^2(\Omega)\) and \(E\) is a Borel subset of \(\Omega\), where \(dV_{2n}\) denotes the classical Lebesgue measure in \(\mathbb C^n\). Moreover if \(u_j,u\in \Phi^k(\Omega)\) and \(u_j\to u\), \(j\to \infty\), locally uniformly in \(\Omega\), then the corresponding sequence of measures \(\mu_k[u_j]\) converges weakly to the measure \(\mu_k[u]\).
In the proof, technics developed by N. S. Trudinger and X.-J. Wang [Topol. Methods Nonlinear Anal. 10, No. 2, 225–239 (1997; Zbl 0915.35039)] are used. A crucial role is played by the monotonicity formula (adapted from the real case): for any \(u,v\in \Phi^k(\Omega)\cap \mathcal C(\overline {\Omega})\) such that \(u\geq v\) in \(\Omega\) and \(u=v\) on \(\partial \Omega\), the function \[ t\mapsto \int_{\Omega}F_k((1-t)u+tv)dV_{2n} \] is nondecreasing for \(t\in [0,1]\).
MSC:
32W20 Complex Monge-Ampère operators
28A33 Spaces of measures, convergence of measures
32U05 Plurisubharmonic functions and generalizations
31C10 Pluriharmonic and plurisubharmonic functions
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