Convergence of multipole Green functions

TL;DR

This paper investigates the convergence behavior of multipole Green functions in complex analysis, establishing conditions under which weak convergence implies stronger forms of convergence and characterizing their limits.

Contribution

It extends the understanding of Green function convergence when poles collide, especially when poles approach different points or the boundary, and reduces the problem to simpler cases.

Findings

Weak convergence implies convergence in capacity.

Convergence is uniform away from poles not approaching the boundary.

Limits can be expressed as maxima of logarithms of holomorphic functions.

Abstract

We continue the study of convergence of multipole pluricomplex Green functions for a bounded hyperconvex domain of $\mathbb C^n$, in the case where poles collide. We consider the case where all poles do not converge to the same point in the domain, and some of them might go to the boundary of the domain. We prove that weak convergence will imply convergence in capacity; that it implies convergence uniformly on compacta away from the poles when no poles tend to the boundary; and that the study can be reduced, in a sense, to the case where poles tend to a single point. Furthermore, we prove that the limits of Green functions can be obtained as limits of functions of the type $\max_{1\le i\le 3n} \frac{1}{p} \log |f_i|$, where the $f_i$ are holomorphic functions.