Delta-extremal Functions in Cn

Abstract

The article is devoted to properties of a weighted Green function. We study the (δ, ψ)- extremal Green function V ∗ δ (z,K, ψ) defined by the class Lδ = { u(z) ∈ psh(Cn) : u(z) 6 Cu + δ ln+ |z|, z ∈ Cn} , δ > 0. We see that the notion of regularity of points with respect to different numbers δ differ from each other. Nevertheless, we prove that if a compact set K ⊂ Cn is regular, then δ-extremal function is continuous in the whole space Cn