Boundary cross theorem in dimension 1 with 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 $((D\cup A)\times B)\cup (A\times (B\cup G)),$ 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" $\widehat{W\setminus M}$ 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 $((A\times G)\cup (D\times B))\setminus M$ "extends" to a function holomorphic on $\widehat{W\setminus M}.$

Preprint of the ICTP, Trieste-Italy (2007). 18 pages

References in corpus (1)

• A unified approach to the theory of separately holomorphic mappings

Cited by in corpus (1)

• A unified approach to the theory of separately holomorphic mappings