Convergence in Capacity

Xing, Yang

Annales de l'Institut Fourier, Tome 58 (2008) no. 5, pp. 1839-1861.

Nous étudions la relation entre la convergence en capacité des fonctions pluri sous-harmoniques et la convergence des mesures de Monge-Ampère complexes correspondantes. Nous trouvons un type de convergence des mesures de Monge-Ampère complexe qui est essentiellement équivalent à la convergence en capacité $C_n$ des fonctions. Nous montrons aussi que la convergence faible des mesures de Monge-Ampère complexes est équivalente à la convergence en capacité $C_{n-1}$ des fonctions dans certains cas. Comme application nous donnons des théorèmes de stabilité des solutions des équations de Monge-Ampère.

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.

Classification : 32W20, 32U15
Mots clés : the complex Monge-Ampère operator, plurisubharmonic function, capacity

[1] Åhag, P. The complex Monge-Ampère operator on bounded hyperconvex domains (2002) (Doctoral thesis, Mid Sweden University, Sundsvall)

[2] Bedford, E.; Taylor, B. A. The Dirichlet problem for the complex Monge-Ampère operator, Invent. Math., Volume 37 (1976), pp. 1-44 | DOI | MR | Zbl

[3] Bedford, E.; Taylor, B. A. A new capacity for plurisubharmonic functions, Acta Math., Volume 149 (1982), pp. 1-40 | DOI | MR | Zbl

[4] Bedford, E.; Taylor, B. A. Fine topology, Šilov boundary and ${\left(d{d}^c\right)}^n$, J. Funct. Anal., Volume 72 (1987), pp. 225-251 | DOI | MR | Zbl

[5] Cegrell, U. Discontinuité de l’opérateur de Monge-Ampère complexe, C. R. Acad. Sci. Paris Ser. I Math., Volume 296 (1983), pp. 869-871 | Zbl

[6] Cegrell, U. Capacities in complex analysis, 1988 (Braunschweig/Wiesbaden:Friedr. Vieweg and Sohn) | MR | Zbl

[7] Cegrell, U. Pluricomplex energy, Acta Math., Volume 180 (1998) no. 2, pp. 187-217 | DOI | MR | Zbl

[8] Cegrell, U. Convergence in capacity, Isaac Newton Institute for Math. Science P., 2001 (reprint Series NI01046-NPD, also available at arxiv.org: math. CV/0505218)

[9] Cegrell, U. The general definition of the complex Monge-Ampère operator, Ann. Inst. Fourier, Volume 54 (2004), pp. 159-179 | DOI | Numdam | MR | Zbl

[10] Cegrell, U.; Kolodziej, S. The equation of complex Monge-Ampère type and stability of solutions, Math. Ann., Volume 334 (2006), pp. 713-729 | DOI | MR | Zbl

[11] Kiselman, C. O. Plurisubharmonic functions and potential theory in several complex variables, Development of mathematics 1950-2000 (Birkhäuser, Basel, 2000, 655-714) | MR | Zbl

[12] Kolodziej, S. The range of the complex Monge-Ampère operator, II, Indiana Univ. Math. J., Volume 44 (1995), pp. 765-782 | DOI | MR | Zbl

[13] Kolodziej, S. The complex Monge-Ampère equation and pluripotential theory, Memoirs of the Amer. Math. Soc., Volume 178 (2005) no. 840, pp. 64p. | MR | Zbl

[14] Lelong, P. Discontinuité et annulation de l’opérateur de Monge-Ampère complexe, Lecture Notes in Math., Volume 1028 (1983), pp. 219-224 (Springer-Verlag, Berlin) | DOI | Zbl

[15] Rashkovskii, A. Singularities of plurisubharmonic functions and positive closed currents, Mid Sweden University, Research Reports, 2000 (No 20)

[16] Xing, Y. Weak Convergence of Currents (To appear in Math. Z.) | Zbl

[17] Xing, Y. Continuity of the complex Monge-Ampère operator, Proc. of Amer. Math. Soc., Volume 124 (1996), pp. 457-467 | DOI | MR | Zbl

[18] Xing, Y. Complex Monge-Ampère measures of plurisubharmonic functions with bounded values near the boundary, Canad. J. Math., Volume 52 (2000) no. 5, pp. 1085-1100 | DOI | MR | Zbl