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Robin functions and extremal functions

T. Bloom, N. Levenberg, S. Ma'u (2003)

Annales Polonici Mathematici

Given a compact set K ⊂ ℂ N , for each positive integer n, let V ( n ) ( z ) = V K ( n ) ( z ) := sup 1 / ( d e g p ) V p ( K ) ( p ( z ) ) : p holomorphic polynomial, 1 ≤ deg p ≤ n. These “extremal-like” functions V K ( n ) are essentially one-variable in nature and always increase to the “true” several-variable (Siciak) extremal function, V K ( z ) := max[0, sup1/(deg p) log|p(z)|: p holomorphic polynomial, | | p | | K ≤ 1 ]. Our main result is that if K is regular, then all of the functions V K ( n ) are continuous; and their associated Robin functions ϱ V K ( n ) ( z ) : = l i m s u p | λ | → ∞ [ V K ( n ) ( λ z ) - l o g ( | λ | ) ] increase to ϱ K : = ϱ V K for all z outside a pluripolar set....

Separately superharmonic functions in product networks

Victor Anandam (2015)

Annales Polonici Mathematici

Let X×Y be the Cartesian product of two locally finite, connected networks that need not have reversible conductance. If X,Y represent random walks, it is known that if X×Y is recurrent, then X,Y are both recurrent. This fact is proved here by non-probabilistic methods, by using the properties of separately superharmonic functions. For this class of functions on the product network X×Y, the Dirichlet solution, balayage, minimum principle etc. are obtained. A unique integral representation is given...

Singular functions on metric measure spaces.

Ilkka Holopainen, Nageswari Shanmugalingam (2002)

Collectanea Mathematica

On relatively compact domains in metric measure spaces we construct singular functions that play the role of Green functions of the p-Laplacian. We give a characterization of metric spaces that support a global version of such singular function, in terms of capacity estimates at infinity of such metric spaces. In addition, when the measure of the space is locally Q-regular, we study quasiconformal invariance property associated with the existence of global singular functions.

Singular sets of separately analytic functions

Zbigniew Błocki (1992)

Annales Polonici Mathematici

We complete the characterization of singular sets of separately analytic functions. In the case of functions of two variables this was earlier done by J. Saint Raymond and J. Siciak.

Singularité et intégrabilité des fonctions plurisousharmoniques

Mongi Blel, Saoud K. Mimouni (2005)

Annales de l’institut Fourier

On étudie les singularités et l’intégrabilité d’une classe de fonctions plurisousharmoniques sur une variété analytique X de dimension n ≥ 1 . Pour étudier ce problème, nous commençons par contrôler les nombres de Lelong de certains types de fonctions plurisousharmoniques ϕ . Ensuite, nous étudions les singularités du transformé strict du courant d d c ϕ par un éclatement de X au dessus d’un point. Nous répondons ainsi positivement au problème d’intégrabilité locale de e - ϕ , lorsque dim X = 2 , et lorsque ϕ est une fonction plurisousharmonique...

Currently displaying 441 – 460 of 622