English

Convergence of multipole Green functions

Complex Variables 2017-10-24 v2

Abstract

We continue the study of convergence of multipole pluricomplex Green functions for a bounded hyperconvex domain of Cn\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 max1i3n1plogfi\max_{1\le i\le 3n} \frac{1}{p} \log |f_i|, where the fif_i are holomorphic functions.

Keywords

Cite

@article{arxiv.1407.0417,
  title  = {Convergence of multipole Green functions},
  author = {Nguyen Quand Dieu and Pascal J. Thomas},
  journal= {arXiv preprint arXiv:1407.0417},
  year   = {2017}
}

Comments

To appear in Indiana University Mathematics Journal

R2 v1 2026-06-22T04:52:59.182Z