Convergence in Capacity

Yang Xing[1]

  • [1] Swedish University of Agricultural Sciences Centre of Biostochastics 901 83 Umeå(Sweden)

Annales de l’institut Fourier (2008)

  • Volume: 58, Issue: 5, page 1839-1861
  • ISSN: 0373-0956

Abstract

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We study the relationship between convergence in capacities of plurisubharmonic functions and the convergence of the corresponding complex Monge-Ampère measures. We find one type of convergence of complex Monge-Ampère measures which is essentially equivalent to convergence in the capacity C n of functions. We also prove that weak convergence of complex Monge-Ampère measures is equivalent to convergence in the capacity C n - 1 of functions in some case. As applications we give certain stability theorems of solutions of Monge-Ampère equations.

How to cite

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Xing, Yang. "Convergence in Capacity." Annales de l’institut Fourier 58.5 (2008): 1839-1861. <http://eudml.org/doc/10364>.

@article{Xing2008,
abstract = {We study the relationship between convergence in capacities of plurisubharmonic functions and the convergence of the corresponding complex Monge-Ampère measures. We find one type of convergence of complex Monge-Ampère measures which is essentially equivalent to convergence in the capacity $C_n$ of functions. We also prove that weak convergence of complex Monge-Ampère measures is equivalent to convergence in the capacity $C_\{n-1\}$ of functions in some case. As applications we give certain stability theorems of solutions of Monge-Ampère equations.},
affiliation = {Swedish University of Agricultural Sciences Centre of Biostochastics 901 83 Umeå(Sweden)},
author = {Xing, Yang},
journal = {Annales de l’institut Fourier},
keywords = {the complex Monge-Ampère operator; plurisubharmonic function; capacity; plurisubharmonic functions; Monge-Ampère operator; capacities},
language = {eng},
number = {5},
pages = {1839-1861},
publisher = {Association des Annales de l’institut Fourier},
title = {Convergence in Capacity},
url = {http://eudml.org/doc/10364},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Xing, Yang
TI - Convergence in Capacity
JO - Annales de l’institut Fourier
PY - 2008
PB - Association des Annales de l’institut Fourier
VL - 58
IS - 5
SP - 1839
EP - 1861
AB - We study the relationship between convergence in capacities of plurisubharmonic functions and the convergence of the corresponding complex Monge-Ampère measures. We find one type of convergence of complex Monge-Ampère measures which is essentially equivalent to convergence in the capacity $C_n$ of functions. We also prove that weak convergence of complex Monge-Ampère measures is equivalent to convergence in the capacity $C_{n-1}$ of functions in some case. As applications we give certain stability theorems of solutions of Monge-Ampère equations.
LA - eng
KW - the complex Monge-Ampère operator; plurisubharmonic function; capacity; plurisubharmonic functions; Monge-Ampère operator; capacities
UR - http://eudml.org/doc/10364
ER -

References

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  1. P. Åhag, The complex Monge-Ampère operator on bounded hyperconvex domains, (2002) 
  2. E. Bedford, B. A. Taylor, The Dirichlet problem for the complex Monge-Ampère operator, Invent. Math. 37 (1976), 1-44 Zbl0315.31007MR445006
  3. E. Bedford, B. A. Taylor, A new capacity for plurisubharmonic functions, Acta Math. 149 (1982), 1-40 Zbl0547.32012MR674165
  4. E. Bedford, B. A. Taylor, Fine topology, Šilov boundary and ( d d c ) n , J. Funct. Anal. 72 (1987), 225-251 Zbl0677.31005MR886812
  5. U. Cegrell, Discontinuité de l’opérateur de Monge-Ampère complexe, C. R. Acad. Sci. Paris Ser. I Math. 296 (1983), 869-871 Zbl0541.31008
  6. U. Cegrell, Capacities in complex analysis, (1988) Zbl0655.32001MR964469
  7. U. Cegrell, Pluricomplex energy, Acta Math. 180 (1998), 187-217 Zbl0926.32042MR1638768
  8. U. Cegrell, Convergence in capacity, Isaac Newton Institute for Math. Science P. (2001) Zbl1251.32028
  9. U. Cegrell, The general definition of the complex Monge-Ampère operator, Ann. Inst. Fourier 54 (2004), 159-179 Zbl1065.32020MR2069125
  10. U. Cegrell, S. Kolodziej, The equation of complex Monge-Ampère type and stability of solutions, Math. Ann. 334 (2006), 713-729 Zbl1103.32019MR2209253
  11. C. O. Kiselman, Plurisubharmonic functions and potential theory in several complex variables, Development of mathematics 1950-2000 Zbl0962.31001MR1796855
  12. S. Kolodziej, The range of the complex Monge-Ampère operator, II, Indiana Univ. Math. J. 44 (1995), 765-782 Zbl0849.31009MR1375348
  13. S. Kolodziej, The complex Monge-Ampère equation and pluripotential theory, Memoirs of the Amer. Math. Soc. 178 (2005) Zbl1084.32027MR2172891
  14. P. Lelong, Discontinuité et annulation de l’opérateur de Monge-Ampère complexe, Lecture Notes in Math. 1028 (1983), 219-224 Zbl0592.32014
  15. A. Rashkovskii, Singularities of plurisubharmonic functions and positive closed currents, Mid Sweden University, Research Reports (2000) Zbl0972.32024
  16. Y. Xing, Weak Convergence of Currents Zbl1165.32016
  17. Y. Xing, Continuity of the complex Monge-Ampère operator, Proc. of Amer. Math. Soc. 124 (1996), 457-467 Zbl0849.31010MR1322940
  18. Y. Xing, Complex Monge-Ampère measures of plurisubharmonic functions with bounded values near the boundary, Canad. J. Math. 52 (2000), 1085-1100 Zbl0976.32019MR1782339

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