Transactions of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

A comparison principle for the complex Monge-Ampère operator in Cegrell's classes and applications

Author(s): Nguyen Van Khue; Pham Hoang Hiep
Journal: Trans. Amer. Math. Soc. 361 (2009), 5539-5554.
MSC (2000): Primary 32W20; Secondary 32U15
Posted: May 15, 2009
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: In this article we will first prove a result about the convergence in capacity. Next we will obtain a general decomposition theorem for complex Monge-Ampère measures which will be used to prove a comparison principle for the complex Monge-Ampère operator.

References:

1.: P. Åhag, The complex Monge-Ampère operator on bounded hyperconvex domains, Ph.D. Thesis, Umeå University, 2002.
2.: Z. Błocki, On the definition of the Monge-Ampère operator in $\mathbf C^2$ , Math. Ann., 328 (2004), 415-423. MR 2036329 (2004m:32074)
3.: Z. Błocki, The domain of definition of the complex Monge-Ampère operator, Amer. J. Math, 128 (2006), 519-530. MR 2214901 (2006k:32076)
4.: E. Bedford and B. A.Taylor, The Dirichlet problem for the complex Monge-Ampère operator, Invent. Math. 37 (1976), 1-44. MR 0445006 (56:3351)
5.: E. Bedford and B. A.Taylor, A new capacity for plurisubharmonic functions, Acta Math., 149 (1982), 1-40. MR 674165 (84d:32024)
6.: E. Bedford and B. A.Taylor, Fine topology, Silov boundary, and , J. Funct. Anal. 72 (1987), 225-251. MR 886812 (88g:32033)
7.: U. Cegrell, Pluricomplex energy, Acta Math., 180 (1998), 187-217. MR 1638768 (99h:32016)
8.: U. Cegrell, The general definition of the complex Monge-Ampère operator, Ann. Inst. Fourier (Grenoble) 54 (2004), 159-179. MR 2069125 (2005d:32062)
9.: U. Cegrell, A general Dirichlet problem for the complex Monge-Ampère operator, Ann. Polon. Math., 94 (2008), 131-147. MR 2438854
10.: U. Cegrell, S. Kołodziej and A. Zeriahi, Subextention of plurisubharmonic functions with weak singularities, Math. Zeit., 250 (2005), 7-22. MR 2136402 (2005m:32064)
11.: R. Czyz, Convergence in capacity of the Perron-Bremermann envelope, Michigan Math. J., 53 (2005), 497-509. MR 2207203 (2006k:32066)
12.: D. Coman, N. Levenberg and E. A. Poletsky, Quasianalyticity and pluripolarity, J. Amer. Math. Soc., 18 (2005), 239-252. MR 2137977 (2006e:32043)
13.: J-P. Demailly, Monge-Ampère operators, Lelong numbers and intersection theory, Complex Analysis and Geometry, Univ. Ser. Math., Plenum, New York, 1993, 115-193. MR 1211880 (94k:32009)
14.: J-P. Demailly, Potential theory in several variables, preprint (1989).
15.: S. Kołodziej, The range of the complex Monge-Ampère operator, II, Indiana Univ. Math. J., 44 (1995), 765-782. MR 1375348 (96m:32013)
16.: P. Hiep, A characterization of bounded plurisubharmonic functions, Ann. Polon. Math., 85 (2004), 233-238. MR 2181753 (2006g:32051)
17.: P. Hiep, The comparison principle and Dirichlet problem in the class $\mathcal E_p(f)$ , , Ann. Polon. Math., 88 (2006), 247-261. MR 2260404 (2007f:32048)
18.: P. Lelong, Notions capacitaires et fonctions de Green pluricomplexes dans les espaces de Banach. C.R. Acad. Sci. Paris Ser. Imath., 305:71-76, 1987. MR 901138 (89g:46082)
19.: Y. Xing, Continuity of the complex Monge-Ampère operator. Proc. Amer. Math. Soc., 124 (1996), 457-467. MR 1322940 (96d:32015)
20.: Y. Xing, Complex Monge-Ampère measures of pluriharmonic functions with bounded values near the boundary. Canad. J. Math., 52, (2000),1085-1100. MR 1782339 (2001h:32070)

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|