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Poletsky has introduced a notion of plurisubharmonicity for functions defined on compact sets in ℂⁿ. We show that these functions can be completely characterized in terms of monotone convergence of plurisubharmonic functions defined on neighborhoods of the compact.

Plurisubharmonic functions on smooth domains.

John Erik Fornaess (1983)

Mathematica Scandinavica

Plurisubharmonic functions with logarithmic singularities

E. Bedford, B. A. Taylor (1988)

Annales de l'institut Fourier

To a plurisubharmonic function $u$ on $C^{n}$ with logarithmic growth at infinity, we may associate the Robin function $ρ_{u} (z) = \underset{λ \to \infty}{lim sup} u (λ z) - log (λ z)$ defined on $P^{n - 1}$ , the hyperplane at infinity. We study the classes $L_{+}$ , and (respectively) $L_{p}$ of plurisubharmonic functions which have the form $u = log (1 + | z |) + O (1)$ and (respectively) for which the function $ρ_{u}$ is not identically $- \infty$ . We obtain an integral formula which connects the Monge-Ampère measure on the space $C^{n}$ with the Robin function on $P^{n - 1}$ . As an application we obtain a criterion on the convergence of the Monge-Ampère...

Plurisubharmonic saddles

Siegfried Momm (1996)

Annales Polonici Mathematici

A certain linear growth of the pluricomplex Green function of a bounded convex domain of $ℂ^{N}$ at a given boundary point is related to the existence of a certain plurisubharmonic function called a “plurisubharmonic saddle”. In view of classical results on the existence of angular derivatives of conformal mappings, for the case of a single complex variable, this allows us to deduce a criterion for the existence of subharmonic saddles.

Polynomial diffeomorphisms of C2: currents, equilibrium measure and hyperbolicity.

Eric Bedford, John Smillie (1991)

Inventiones mathematicae

Prolongement d’un courant positif quasi-plurisurharmonique

Noureddine Ghiloufi, Khalifa Dabbek (2009)

Annales mathématiques Blaise Pascal

Le but de cet article est de montrer un résultat de prolongement d’un courant positif, défini en dehors d’un obstacle fermé, dont le $d d^{c}$ est dominé par un courant positif fermé de masse localement finie. On étudie divers types d’obstacles : soit un ensemble fermé pluripolaire complet, soit l’ensemble des zéros d’une fonction strictement $k$ -convexe positive. Dans la troisième partie, sous des conditions sur la dimension de Hausdorff de l’obstacle, on démontre le prolongement d’un tel courant. On termine...

Pseudoconvex domains over $q$ -complete manifolds

Viorel Vâjâitu (2000)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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