# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
A new capacity for plurisubharmonic functions. (English) Zbl 0547.32012
This is an important paper in which the complex Monge-Ampère operator ${\left(d{d}^{c}\right)}^{n}$ 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$\circ \right)$ continuity of the operator ${\left(d{d}^{c}\right)}^{k}$ (1$\le k\le n\right)$ under decreasing limits; (2$\circ \right)$ quasicontinuity of plurisubharmonic function with respect to the relative capacity $c\left(K,{\Omega }\right):=sup\left\{{\int }_{K}{\left(d{d}^{c}v\right)}^{n};\phantom{\rule{1.em}{0ex}}v\in PSH\left({\Omega }\right),\phantom{\rule{1.em}{0ex}}0 where ${\Omega }\subset {ℂ}^{n}$ is a fixed open set and $K\subset {\Omega }$ is compact; (3$\circ \right)$ domination principle for plurisubharmonic functions; (4$\circ \right)$ 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 Generalizations of potentials and capacities 31C10 Pluriharmonic and plurisubharmonic functions
##### 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 · doi:10.1007/BF01363896 [3] – Envelopes of continuous, plurisubharmonic functions.Math. Ann., 251 (1980), 175–183. · Zbl 0433.31012 · doi:10.1007/BF01536184 [4] Bedford, E. &Taylor, B. A., The Dirichlet problem for the complex Monge-Ampère equation.Invent. Math. 37 (1976), 1–44. · Zbl 0325.31013 · doi:10.1007/BF01418826 [5] –, Variational properties of the complex Monge-Ampère equation, I: Dirichlet Principle.Duke Math. J., 45 (1978), 375–403. · Zbl 0401.35093 · doi:10.1215/S0012-7094-78-04520-9 [6] –, Variational properties of the complex Monge-Ampère equation, II: Intrinsic norms.Amer. J. Math., 101 (1979), 1131–1166. · Zbl 0446.35025 · doi:10.2307/2374130 [7] Brelot, M.,Elements de la théorie classique du potentiel. Centre de Documentation, Paris, 1959. [8] Cartan, H., Capacité extérieure et suites convergentes de potentiels.C.R. Acad. Sci. Paris, (1942), 944–946. [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. [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. [14] –, Sui les points d’effilement d’un ensemble. Application a l’étude de la capacité.Ann. Inst. Fourier, 9 (1959), 91–101. [15] Federer, H.,Geometric Measure Theory. Springer-Verlag, New York, 1969. [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 · doi:10.1016/0022-1236(77)90046-5 [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 · doi:10.1007/BF02385986 [18] Lelong, P.,Plurisubharmonic functions and positive differential forms. Gordon and Breach, New York, 1969. [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. [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 · doi:10.1007/BF01450654 [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. [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 · doi:10.1070/SM1980v036n04ABEH001861 [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 · doi:10.2307/1971038 [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.