Hostname: page-component-5b777bbd6c-f9nfp Total loading time: 0 Render date: 2025-06-18T09:15:53.866Z Has data issue: false hasContentIssue false
  • English
  • Français

WEAK CONTINUITY OF THE COMPLEX $k$-HESSIAN OPERATORS WITH RESPECT TO LOCAL UNIFORM CONVERGENCE

Part of: Classical measure theory Differential operators in several variables Other generalizations Pluripotential theory

Published online by Cambridge University Press:  11 June 2013

NEIL S. TRUDINGER  and
WEI ZHANG
NEIL S. TRUDINGER*
Affiliation:
Centre for Mathematics and its Applications, The Australian National University, Canberra, ACT 0200, Australia email wei.zhang@anu.edu.au
WEI ZHANG
Affiliation:
Centre for Mathematics and its Applications, The Australian National University, Canberra, ACT 0200, Australia email wei.zhang@anu.edu.au
*
✉neil.trudinger@anu.edu.au
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we study the properties of $k$-plurisubharmonic functions defined on domains in ${ \mathbb{C} }^{n} $. By the monotonicity formula, we give an alternative proof of the weak continuity of complex $k$-Hessian operators with respect to local uniform convergence.

Keywords

complex Monge–Ampère operatorcomplex k-Hessian measuresweak convergencemonotonicity formula

MSC classification

Secondary: 32W20: Complex Monge-Ampère operators 28A33: Spaces of measures, convergence of measures 32U05: Plurisubharmonic functions and generalizations 31C10: Pluriharmonic and plurisubharmonic functions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 89 , Issue 2 , April 2014 , pp. 227 - 233
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

References

Bedford, E. and Taylor, B. A., ‘The Dirichlet problem for a complex Monge–Ampère equation’, Invent. Math. 37 (1) (1976), 1–44.CrossRefGoogle Scholar
Bedford, E. and Taylor, B. A., ‘A new capacity for plurisubharmonic functions’, Acta Math. 149 (1–2) (1982), 1–40.Google Scholar
Bedford, E. and Taylor, B. A., ‘Fine topology, Šilov boundary, and $\mathop{(d{d}^{c} )}\nolimits ^{n} $’, J. Funct. Anal. 72 (2) (1987), 225–251.Google Scholar
Blocki, Z., ‘The domain of definition of the complex Monge–Ampère operator’, Amer. J. Math. 128 (2) (2006), 519–530.Google Scholar
Cegrell, U., ‘Discontinuité de l’opérateur de Monge–Ampère complexe’, C. R. Acad. Sci. Paris Sér. I Math. 296 (21) (1983), 869–871.Google Scholar
Cegrell, U., ‘Sums of continuous plurisubharmonic functions and the complex Monge–Ampère operator in ${ \mathbb{C} }^{n} $’, Math. Z. 193 (3) (1986), 373–380.CrossRefGoogle Scholar
Chern, S. S., Levine, H. I. and Nirenberg, L., ‘Intrinsic norms on a complex manifold’, in: Global Analysis (Papers in Honor of K. Kodaira) (University Tokyo Press, Tokyo, 1969), 119–139.Google Scholar
Demailly, J.-P., ‘Mesures de Monge–Ampère et caractérisation géométrique des variétés algébriques affines’, Mém. Soc. Math. France (N.S.) (19) (1985).Google Scholar
Garofalo, N. and Tournier, F., ‘New properties of convex functions in the Heisenberg group’, Trans. Amer. Math. Soc. 358 (5) (2006), 2011–2055.Google Scholar
Guedj, V. and Zeriahi, A., ‘The weighted Monge–Ampère energy of quasiplurisubharmonic functions’, J. Funct. Anal. 250 (2) (2007), 442–482.CrossRefGoogle Scholar
Gutiérrez, C. E. and Montanari, A., ‘Maximum and comparison principles for convex functions on the Heisenberg group’, Comm. Partial Differential Equations 29 (9–10) (2004), 1305–1334.Google Scholar
Gutiérrez, C. E. and Montanari, A., ‘On the second order derivatives of convex functions on the Heisenberg group’, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 3 (2) (2004), 349–366.Google Scholar
Kiselman, C. O., ‘Sur la définition de l’opérateur de Monge–Ampère complexe’, in: Complex Analysis (Toulouse, 1983), Lecture Notes in Mathematics, 1094 (Springer, Berlin, 1984), 139–150.Google Scholar
Reilly, R. C., ‘On the Hessian of a function and the curvatures of its graph’, Michigan Math. J. 20 (1973), 373–383.Google Scholar
Sibony, N., ‘Quelques problèmes de prolongement de courants en analyse complexe’, Duke Math. J. 52 (1) (1985), 157–197.CrossRefGoogle Scholar
Trudinger, N. S. and Wang, X.-J., ‘Hessian measures. I’, Topol. Methods Nonlinear Anal. 10 (2) (1997), 225–239.Google Scholar
Trudinger, N. S. and Wang, X.-J., ‘Hessian measures. II’, Ann. of Math. (2) 150 (2) (1999), 579–604.CrossRefGoogle Scholar
Trudinger, N. S. and Zhang, W., ‘Hessian measures on the Heisenberg group’, J. Funct. Anal. 264 (10) (2013), 2335–2355.Google Scholar
Xing, Y., ‘Continuity of the complex Monge–Ampère operator’, Proc. Amer. Math. Soc. 124 (2) (1996), 457–467.Google Scholar