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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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