Boundary cross theorem in dimension 1

Abstract

Let $X,Y$ be two complex manifolds of dimension 1 which are countable at infinity, let $D\subset X,$ $G\subset Y$ be two open sets, let $A$ (resp. $B$) be a subset of $\partial D$ (resp. $\partial G$), and let $W$ be the 2-fold cross $((D\cup A)\times B)\cup (A\times (B\cup G)).$ Suppose in addition that $D$ (resp. $G$) is {\it Jordan-curve-like on $A$} (resp. $B$) and that $A$ and $B$ are {\it of positive length}. We determine the "envelope of holomorphy" $\hat{W}$ of $W$ in the sense that any function locally bounded on $W,$ measurable on $A\times B,$ and separately holomorphic on $(A\times G)\cup (D\times B)$ "extends" to a function holomorphic on the interior of $\hat{W}.$

43 pages, to appear in "Annales Polonici Mathematici". This is the revised version of our article put on Arxiv in March 2005