zbMATH — the first resource for mathematics

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 \(t\in [0,1]\).
32W20 Complex Monge-Ampère operators
28A33 Spaces of measures, convergence of measures
32U05 Plurisubharmonic functions and generalizations
31C10 Pluriharmonic and plurisubharmonic functions
Full Text: DOI
[1] DOI: 10.1090/S0002-9939-96-03316-3 · Zbl 0849.31010 · doi:10.1090/S0002-9939-96-03316-3
[2] DOI: 10.1007/BF01418826 · Zbl 0315.31007 · doi:10.1007/BF01418826
[3] DOI: 10.2307/121089 · Zbl 0947.35055 · doi:10.2307/121089
[4] DOI: 10.1090/S0002-9947-05-04016-X · Zbl 1102.35033 · doi:10.1090/S0002-9947-05-04016-X
[5] Trudinger, Topol. Methods Nonlinear Anal. 10 pp 225– (1997) · Zbl 0915.35039 · doi:10.12775/TMNA.1997.030
[6] Demailly, Mém. Soc. Math. France (N.S.) (1985)
[7] DOI: 10.1215/S0012-7094-85-05210-X · Zbl 0578.32023 · doi:10.1215/S0012-7094-85-05210-X
[8] Chern, Global Analysis (Papers in Honor of K. Kodaira) pp 119– (1969)
[9] Reilly, Michigan Math. J. 20 pp 373– (1973)
[10] DOI: 10.1007/BF01229804 · Zbl 0624.31004 · doi:10.1007/BF01229804
[11] Kiselman, Complex Analysis (Toulouse, 1983) pp 139– (1984) · Zbl 0585.32019
[12] Gutiérrez, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 3 pp 349– (2004)
[13] Cegrell, C. R. Acad. Sci. Paris Sér. I Math. 296 pp 869– (1983)
[14] DOI: 10.1081/PDE-200037752 · Zbl 1056.35033 · doi:10.1081/PDE-200037752
[15] DOI: 10.1353/ajm.2006.0010 · Zbl 1102.32018 · doi:10.1353/ajm.2006.0010
[16] DOI: 10.1016/j.jfa.2007.04.018 · Zbl 1143.32022 · doi:10.1016/j.jfa.2007.04.018
[17] DOI: 10.1016/0022-1236(87)90087-5 · Zbl 0677.31005 · doi:10.1016/0022-1236(87)90087-5
[18] DOI: 10.1007/BF02392348 · doi:10.1007/BF02392348
[19] DOI: 10.1016/j.jfa.2013.02.019 · Zbl 1282.26016 · doi:10.1016/j.jfa.2013.02.019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.