Extension of separately analytic functions with thin singularities on one-dimensional parallel sections

Abstract

Let two bounded simply connected domains $D\subset {\mathbb C}_{z}$, $G\subset {\mathbb C}_{w}$, and two locally regular compact subsets $E\subset D,\, \, \, F\subset G$ be given. If $f(z,w)$ is a separately analytic function with finitely many singular points in each section of $D\times \left\{w^{0} \right\}$ and $\left\{z^{0} \right\}\times G$ for any $(z^{0} ,w^{0} )\in E\times F$ on the set $X=(D\times F)\cup (E\times G),$ then it extends holomorphically to the domain \[\hat{X}=\left\{(z,w)\in D\times G:\, \, \, \omega^{*}(z,E,D)+\omega^{*}(w,F,G)<-1\right\},\] except on an analytic set $S$. Where $\omega^*(z,E,D)$ is the harmonic measure of the set $E\subset D\subset {\mathbb C}$ relative to the domain $D$.