Boundary cross theorem in dimension 1 with singularities

TL;DR

This paper extends the boundary cross theorem to the case of singularities in one complex dimension, characterizing the envelope of holomorphy for functions with certain boundary and singularity conditions.

Contribution

It determines the envelope of holomorphy for boundary crosses with fiberwise polar or discrete singularities, generalizing classical results to include singularities.

Findings

Explicit description of the envelope of holomorphy for the given boundary cross.

Extension of functions with boundary measurability and separate holomorphicity across singularities.

Generalization of boundary cross theorems to include fiberwise polar and discrete singularities.

Abstract

Let $D$ and $G$ be copies of the open unit disc in $\C,$ let $A$ (resp. $B$) be a measurable subset of $\partial D$ (resp. $\partial G$), let $W$ be the 2-fold cross $\big((D\cup A)\times B\big)\cup \big(A\times(B\cup G)\big),$ and let $M$ be a relatively closed subset of $W.$ Suppose in addition that $A$ and $B$ are of positive one-dimensional Lebesgue measure and that $M$ is fiberwise polar (resp. fiberwise discrete) and that $M\cap (A\times B)=\varnothing.$ We determine the "envelope of holomorphy" $\hat{W\setminus M}$ of $W\setminus M$ in the sense that any function locally bounded on $W\setminus M,$ measurable on $A\times B,$ and separately holomorphic on $\big((A\times G) \cup (D\times B)\big)\setminus M$ "extends" to a function holomorphic on $\hat{W\setminus M}.$