On the Converse of Shimoda's Theorem

Abstract

We study Shimoda's theorem on the holomorphicity of a function $f(z,w)$ defined in a bidisc $U\times V=\left\{ \left| z \right|<1 \right\}\times \left\{ \left| w \right|<1 \right\},$ which is holomorphic with respect to $w\in V$ for each fixed $z\in U$ and is holomorphic with respect to $z\in U$ for each fixed $w\in E$, where $E\subset V$ is a countable set with at least one limit point in $V$. Shimoda proved that in this case $f(z,w)$ is holomorphic in $U\times V$ except for a nowhere dense closed subset of $U\times V$. We prove a converse: for any relatively closed nowhere dense set $S\subset U$ such that $\partial U\bigcup S$ is connected, there exists a function satisfying the conditions of Shimoda's theorem that is holomorphic in $(U\setminus S)\times V$ and doesn't extend holomorphically to any point of $S\times V$.