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A class of maximal plurisubharmonic functions

Azimbay Sadullaev (2012)

Annales Polonici Mathematici

We consider a class of maximal plurisubharmonic functions and prove several properties of it. We also give a condition of maximality for unbounded plurisubharmonic functions in terms of the Monge-Ampère operator ( d d c e u ) .

A comparative analysis of Bernstein type estimates for the derivative of multivariate polynomials

Szilárd Gy. Révész (2006)

Annales Polonici Mathematici

We compare the yields of two methods to obtain Bernstein type pointwise estimates for the derivative of a multivariate polynomial in a domain where the polynomial is assumed to have sup norm at most 1. One method, due to Sarantopoulos, relies on inscribing ellipses in a convex domain K. The other, pluripotential-theoretic approach, mainly due to Baran, works for even more general sets, and uses the pluricomplex Green function (the Zaharjuta-Siciak extremal function). When the inscribed ellipse method...

A constant in pluripotential theory

Zbigniew Błocki (1992)

Annales Polonici Mathematici

We compute the constant sup ( 1 / d e g P ) ( m a x S l o g | P | - S l o g | P | d σ ) : P a polynomial in n , where S denotes the euclidean unit sphere in n and σ its unitary surface measure.

A differential geometric characterization of invariant domains of holomorphy

Gregor Fels (1995)

Annales de l'institut Fourier

Let G = K be a complex reductive group. We give a description both of domains Ω G and plurisubharmonic functions, which are invariant by the compact group, K , acting on G by (right) translation. This is done in terms of curvature of the associated Riemannian symmetric space M : = G / K . Such an invariant domain Ω with a smooth boundary is Stein if and only if the corresponding domain Ω M M is geodesically convex and the sectional curvature of its boundary S : = Ω M fulfills the condition K S ( E ) K M ( E ) + k ( E , n ) . The term k ( E , n ) is explicitly computable...

A Green's function for θ-incomplete polynomials

Joe Callaghan (2007)

Annales Polonici Mathematici

Let K be any subset of N . We define a pluricomplex Green’s function V K , θ for θ-incomplete polynomials. We establish properties of V K , θ analogous to those of the weighted pluricomplex Green’s function. When K is a regular compact subset of N , we show that every continuous function that can be approximated uniformly on K by θ-incomplete polynomials, must vanish on K s u p p ( d d c V K , θ ) N . We prove a version of Siciak’s theorem and a comparison theorem for θ-incomplete polynomials. We compute s u p p ( d d c V K , θ ) N when K is a compact section.

A Hilbert Lemniscate Theorem in 2

Thomas Bloom, Norman Levenberg, Yu. Lyubarskii (2008)

Annales de l’institut Fourier

For a regular, compact, polynomially convex circled set K in C 2 , we construct a sequence of pairs { P n , Q n } of homogeneous polynomials in two variables with deg P n = deg Q n ...

A Liouville theorem for plurisubharmonic currents

Fredj Elkhadhra, Souad Mimouni (2010)

Annales de la faculté des sciences de Toulouse Mathématiques

The goal of this paper is to extend the concepts of algebraic and Liouville currents, previously defined for positive closed currents by M. Blel, S. Mimouni and G. Raby, to psh currents on n . Thus, we study the growth of the projective mass of positive currents on n whose support is contained in a tubular neighborhood of an algebraic subvariety. We also give a sufficient condition guaranteeing that a negative psh current is Liouville. Moreover, we prove that every negative psh algebraic current...

A new characterization of the analytic surfaces in 3 that satisfy the local Phragmén-Lindelöf condition

Rüdiger W. Braun, Reinhold Meise, B. A. Taylor (2011)

Annales de la faculté des sciences de Toulouse Mathématiques

We prove that an analytic surface V in a neighborhood of the origin in 3 satisfies the local Phragmén-Lindelöf condition PL loc at the origin if and only if V satisfies the following two conditions: (1) V is nearly hyperbolic; (2) for each real simple curve γ in 3 and each d 1 , the (algebraic) limit variety T γ , d V satisfies the strong Phragmén-Lindelöf condition. These conditions are also necessary for any pure k -dimensional analytic variety V to satisify PL loc .

A new class of pluripolar sets

Nguyen Quang Dieu, Tang Van Long (2007)

Annales Polonici Mathematici

Let D be a domain in ℂⁿ. We introduce a class of pluripolar sets in D which is essentially contained in the class of complete pluripolar sets. An application of this new class to the problem of approximation of holomorphic functions is also given.

A note on Rosay's paper

Armen Edigarian (2003)

Annales Polonici Mathematici

We give a simplified proof of J. P. Rosay's result on plurisubharmonicity of the envelope of the Poisson functional [10].

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