## A new capacity for plurisubharmonic functions.(English)Zbl 0547.32012

This is an important paper in which the complex Monge-Ampère operator $$(dd^ c)^ n$$ is used to replace the Laplacian and to prove, in the category of complex analysis, the analogues of some well known results of classical potential theory. Let us mention here the following ones: (1$$\circ)$$ continuity of the operator $$(dd^ c)^ k$$ (1$$\leq k\leq n)$$ under decreasing limits; (2$$\circ)$$ quasicontinuity of plurisubharmonic function with respect to the relative capacity $$c(K,\Omega):=\sup\{\int_{K}(dd^ cv)^ n;\quad v\in PSH(\Omega),\quad 0<v<1\},$$ where $$\Omega\subset {\mathbb{C}}^ n$$ is a fixed open set and $$K\subset\Omega$$ is compact; (3$$\circ)$$ domination principle for plurisubharmonic functions; (4$$\circ)$$ negligible sets are pluripolar (solution of an old problem due to P. Lelong). As a consequence one finds that various capacities considered in complex analysis satisfy the axioms of Choquet capacity and hence Borel sets are capacitable.
Reviewer: J.Siciak

### MSC:

 32U05 Plurisubharmonic functions and generalizations 31C15 Potentials and capacities on other spaces 31C10 Pluriharmonic and plurisubharmonic functions
Full Text:

### References:

 [1] Alexander, H., Projective capacity. InRecent Developments in Several Complex Variables, pp. 3–27. J. E. Fornaess, ed., Princeton Univ. Press, 1981. [2] Bedford, E., Extremal plurisubharmonic functions and pluripolar sets inC 2.Math. Ann., 249 (1980), 205–223. · Zbl 0429.31001 [3] – Envelopes of continuous, plurisubharmonic functions.Math. Ann., 251 (1980), 175–183. · Zbl 0433.31012 [4] Bedford, E. &Taylor, B. A., The Dirichlet problem for the complex Monge-Ampère equation.Invent. Math. 37 (1976), 1–44. · Zbl 0325.31013 [5] –, Variational properties of the complex Monge-Ampère equation, I: Dirichlet Principle.Duke Math. J., 45 (1978), 375–403. · Zbl 0401.35093 [6] –, Variational properties of the complex Monge-Ampère equation, II: Intrinsic norms.Amer. J. Math., 101 (1979), 1131–1166. · Zbl 0446.35025 [7] Brelot, M.,Elements de la théorie classique du potentiel. Centre de Documentation, Paris, 1959. · Zbl 0084.30903 [8] Cartan, H., Capacité extérieure et suites convergentes de potentiels.C.R. Acad. Sci. Paris, (1942), 944–946. · Zbl 0061.22607 [9] –, Theorie du potentiel Newtonien: énérgie, capacité, suites de potentiels.Bull. Soc. Math. France, 73 (1945), 74–106. [10] Cegrell, U., Construction of capacities onC n . Preprint. [11] Chern, S. S., Levine, H. I. &Nirenberg, L., Intrinsic norms on a complex manifold.Global Analysis (Papers in honor of K. Kodaira), pp. 119–139, Univ. of Tokyo Press, Tokyo, 1969. · Zbl 0202.11603 [12] Choquet, G., Theory of capacities,Ann. Inst. Fourier, 5 (1953), 131–292. [13] –, Potentiels sur un ensemble de capacité nulle. Suites de potentiels.C.R. Acad. Sci. Paris, 244 (1957), 1707–1710. · Zbl 0086.30601 [14] –, Sui les points d’effilement d’un ensemble. Application a l’étude de la capacité.Ann. Inst. Fourier, 9 (1959), 91–101. · Zbl 0093.29702 [15] Federer, H.,Geometric Measure Theory. Springer-Verlag, New York, 1969. · Zbl 0176.00801 [16] Gaveau, B., Méthodes de contrôle optimal en analyse complexe, I: Résolution d’équations de Monge-Ampère,J. Funct. Anal., 25 (1977) 391–411. · Zbl 0356.35071 [17] Josefson, B., On the equivalence between locally polar and globally polar sets for plurisubharmonic functions onC n .Ark. Mat., 16 (1978), 109–115. · Zbl 0383.31003 [18] Lelong, P.,Plurisubharmonic functions and positive differential forms. Gordon and Breach, New York, 1969. · Zbl 0195.11604 [19] –,Functionelles analytiques et fonctions entieres (n variables). Univ. of Montreal, Montreal (1968). [20] –, Fonctions plurisousharmoniques et fonctions analytiques de variables réeles.Ann. Inst. Fourier, 16 (1966), 269–318. [21] –, Integration sur en ensemble analytique complexe.Bull. Soc. Math. France, 85 (1957), 239–262. · Zbl 0079.30901 [22] Molzon, R., Shiffman, B. &Sibony, N., Average growth estimates for hyperplane sections of entire analytic sets.Math. Ann., 257 (1981), 43–59. · Zbl 0537.32009 [23] Ronkin, L., Regularization of the supremum of a family of plurisubharmonic functions and its application to analytic functions of several variables,Math. USSR-Sb., 71 (113), (1966), 132–142. · Zbl 0182.10902 [24] Sadullaev, A., The operator (dd c u) n and condenser capacities (Russian).Dokl. Akad. Nauk SSSR, 1980, 44–47. [25] –, Deficient divisors in the Valiron sense.Math. USSR-Sb., 36 (1980), 535–547. · Zbl 0443.32015 [26] Siciak, J., Extremal plurisubharmonic functions inC n .Proc. First Finnish-Polish Summer School in Complex Analysis, 1977, 115–152. [27] Siu, Y. T., Extension of meromorphic maps.Ann. of Math., 102 (1975), 421–462. · Zbl 0318.32007 [28] Zaharjuta, V. P., Extremal plurisubharmonic functions, orthogonal polynomials, and the Bernstein-Walsh theorem for analytic functions of several complex variables.Ann. Polon. Math., 33 (1976), 137–148.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.