Boundary cross theorem in dimension 1

Volume 90 / 2007

Peter Pflug, Viêt-Anh Nguyên Annales Polonici Mathematici 90 (2007), 149-192 MSC: Primary 32D15, 32D10. DOI: 10.4064/ap90-2-5


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 $\widehat{W}.$


  • Peter PflugFachbereich Mathematik
    Carl von Ossietzky Universität Oldenburg
    Postfach 2503
    D-26111 Oldenburg, Germany
  • Viêt-Anh NguyênMathematics Section
    The Abdus Salam International Centre
    for Theoretical Physics
    Strada Costiera, 11
    34014 Trieste, Italy

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