arXiv subjects
Publications and source records attributed to Giuseppe Zampieri.
Let D be a convex domain with smooth boundary in complex space and let f be a continuous function on the boundary of D. Suppose that f holomorphically extends to the extremal discs tangent to a convex subdomain of D. We prove that f holomorphically extends to D. The result partially answers a conjecture by Globevnik and Stout of 1991.
Given a wedge V with generic edge in a submanifold M of $C^N$, in general, CR-functions on V do not extend to a wedge in $C^N$ in the direction of each nontrivial value of the Levi form of M as was recently observed by Eastwood and Graham. In this paper we propose an invariant geometric way of selecting those Levi form directions that are responsible for the extension. These are the Levi form directions of the complex tangent vectors, whose complex planes intersect the tangent cone to the wedge in angles larger than $π/2$. We show by several examples the optimality of this criterion. Applications to regularity of CR-functions are also given generalizing the edge-of-the-wedge theorem of Ajrapetyan-Henkin.
Necessary and sufficient geometric conditions are given for domains with regular boundary points and edges to be domains of holomorphy provided the remainder boundary subset is of zero Hausdorff 1-codimensional measure.