Boundary cross theorem in dimension 1

Peter Pflug¹ ; Viêt-Anh Nguyên²

¹ Fachbereich Mathematik Carl von Ossietzky Universität Oldenburg Postfach 2503 D-26111 Oldenburg, Germany
² Mathematics Section The Abdus Salam International Centre for Theoretical Physics Strada Costiera, 11 34014 Trieste, Italy

Annales Polonici Mathematici, Tome 90 (2007) no. 2, pp. 149-192

Résumé

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 $\widehat{W}.$

Détail

DOI : 10.4064/ap90-2-5

Keywords: complex manifolds dimension which countable infinity subset subset sets resp subset partial resp partial fold cross cup times cup times cup suppose addition resp jordan curve like resp positive length determine envelope holomorphy widehat sense function locally bounded measurable times separately holomorphic times cup times extends function holomorphic interior widehat