Boundary cross theorem in dimension 1
Volume 90 / 2007
Annales Polonici Mathematici 90 (2007), 149-192
MSC: Primary 32D15, 32D10.
DOI: 10.4064/ap90-2-5
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 Jordan-curve-like on $A$ (resp. $B$) and that $A$ and $B$ are of positive length. We determine the “envelope of holomorphy” $\widehat{W}$ 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 $\widehat{W}.$
