Skip to Main Content

Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Continuity of the complex Monge-Ampère operator
HTML articles powered by AMS MathViewer

by Yang Xing PDF
Proc. Amer. Math. Soc. 124 (1996), 457-467 Request permission

Abstract:

The complex Monge-Ampère operator $(dd^c)^n$ is an important tool in complex analysis. It would be interesting to find the right notion of convergence $u_j\to u$ such that $(dd^cu_j)^n\to (dd^cu)^n$ in the weak topology. In this paper, using the $C_{n-1}$-capacity, we give a sufficient condition of the weak convergence $(dd^cu_j)^n\to (dd^cu)^n$. We also show that our condition is quite sharp in some case.
References
  • Eric Bedford and B. A. Taylor, The Dirichlet problem for a complex Monge-Ampère equation, Invent. Math. 37 (1976), no. 1, 1–44. MR 445006, DOI 10.1007/BF01418826
  • Eric Bedford and B. A. Taylor, A new capacity for plurisubharmonic functions, Acta Math. 149 (1982), no. 1-2, 1–40. MR 674165, DOI 10.1007/BF02392348
  • Urban Cegrell, Discontinuité de l’opérateur de Monge-Ampère complexe, C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), no. 21, 869–871 (French, with English summary). MR 715323
  • Urban Cegrell and Azim Sadullaev, Approximation of plurisubharmonic functions and the Dirichlet problem for the complex Monge-Ampère operator, Math. Scand. 71 (1992), no. 1, 62–68. MR 1216103, DOI 10.7146/math.scand.a-12410
  • P. Lelong, Fonctions plurisousharmoniques et formes différentielles positives, Gordon & Breach, Paris-London-New York; distributed by Dunod Éditeur, Paris, 1968 (French). MR 0243112
  • Pierre Lelong, Discontinuité et annulation de l’opérateur de Monge-Ampère complexe, P. Lelong-P. Dolbeault-H. Skoda analysis seminar, 1981/1983, Lecture Notes in Math., vol. 1028, Springer, Berlin, 1983, pp. 219–224 (French). MR 774977, DOI 10.1007/BFb0071683
  • L. I. Ronkin, Weak convergence of the currents $[dd^{c}u_{t}]^{q},$ and asymptotic behavior of order functions for holomorphic mappings of regular growth, Sibirsk. Mat. Zh. 25 (1984), no. 4, 167–173 (Russian). MR 754752
  • Y. Xing, On convergence of the current $(dd^cu_t)^q$, Séminaire P. Lelong-P. Dolbeault- H. Skoda, Lecture Notes in Math. (to appear).
Similar Articles
  • Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 32F07, 32F05
  • Retrieve articles in all journals with MSC (1991): 32F07, 32F05
Additional Information
  • Yang Xing
  • Affiliation: Department of Mathematics, University of Umeå, S-901 87 Umeå, Sweden
  • Email: yang.xing@mathdept.umu.se
  • Received by editor(s): August 15, 1994
  • Additional Notes: Partially supported by the Swedish Natural Science Research Council.
  • Communicated by: Eric Bedford
  • © Copyright 1996 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 124 (1996), 457-467
  • MSC (1991): Primary 32F07; Secondary 32F05
  • DOI: https://doi.org/10.1090/S0002-9939-96-03316-3
  • MathSciNet review: 1322940