Boundary behavior of the pluricomplex Green function

Abstract

Let Ω be a bounded domain in Cⁿ. This paper deals with the study of the behavior of the pluricomplex Green function g_Ω(z, w) when the pole w tends to a boundary point w₀ of Ω. We find conditions on Ω which ensure that lim_w→wog_Ω(z, w)=0, uniformly with respect to z on compact subsets of $\bar \Omega \backslash \{ w_0 \} $ . Our main result is Theorem 5; it gives a sufficient condition for the above property to hold, formulated in terms of the existence of a plurisubharmonic peak function for Ω at w₀ which satisfies a certain growth condition.