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arXiv · 0705.4649

Boundary cross theorem in dimension 1 with singularities

Peter Pflug·Viet-Anh Nguyen
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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 $\hat{W\setminus M}.$

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Peter Pflug, Viet-Anh Nguyen. 2007-05-31. Boundary cross theorem in dimension 1 with singularities. https://arxiv.org/abs/0705.4649

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