Mathematics – Complex Variables
Scientific paper
2007-05-31
Mathematics
Complex Variables
Preprint of the ICTP, Trieste-Italy (2007). 18 pages
Scientific paper
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 $\hat{W\setminus M}.$
Nguyen Viet-Anh
Pflug Peter
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