Vitali's theorem without uniform boundedness

Abstract

Let $\{f_m\}_{m \ge 1}$ be a sequence of holomorphic functions defined on a bounded domain $D \subset \mathbb C^n$ or a sequence of rational functions $(1 \le \deg r_m \le m)$ defined on $\mathbb C^n$. We are interested in finding sufficient conditions to ensure the convergence of $\{f_m\}_{m \ge 1}$ on a large set provided the convergence holds pointwise on a not too small set. This type of result is inspired from a theorem of Vitali which gives a positive answer for uniformly bounded sequence.