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Hartogs domains in C^2 and propagation of continuity
Stockholm University, Faculty of Science, Department of Mathematics.
(English)Manuscript (preprint) (Other academic)
National Category
URN: urn:nbn:se:su:diva-25635OAI: diva2:200103
Available from: 2008-11-27 Created: 2008-11-18 Last updated: 2013-07-09Bibliographically approved
In thesis
1. On Propagation of Boundary Continuity for Domains in Complex Space
Open this publication in new window or tab >>On Propagation of Boundary Continuity for Domains in Complex Space
2008 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

We consider continuity properties of analytic functions in bounded domains in complex space, focusing on the following two questions:

Question 1. Given a bounded domain G in complex affine n-space. For which subsets S of the boundary of G does the following hold? If f is a bounded function which is analytic in G and extends continuously to the union of G and S then f extends continously to the closure of G.

Question 2. Given a domain G as above and a function f analytic in G and continuous in the closure of G. For which subsets S of the boundary is the rate of continuity of f on the closure of G determined already by its rate of continuity along S?

Previously it was known that in some cases, the set S may be taken to be the Shilov boundary of the algebra of analytic functions on the domain which are continuous on the closure of the domain, but examples when this is not sufficient were also known.

In Paper I we prove that for a bounded, smoothly bounded pseudoconvex domain G in 2-dimensional complex affine space one may take for S an open neighbourhood of the Shilov boundary (in the boundary of the domain G). This is joint work with B. Jöricke.

Paper II considers smoothly bounded Reinhardt domains in 2 dimensions and complete Reinhardt domains in complex affine n-space. We prove that in many cases one may let S be the Shilov boundary. Moreover, the respective continuity properties propagate to the closure of the envelope of holomorphy of the domain.

Paper III considers pseudoconvex Hartogs domains with connected vertical sections in two dimensions. We prove that in some cases S may be taken to be the Shilov boundary and give examples for when this is not enough.

Place, publisher, year, edition, pages
Stockholm: Matematiska institutionen, 2008. 36 p.
Boundary behavior of holomorphic functions, Global boundary behavior of holomorphic functions
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Research subject
urn:nbn:se:su:diva-8334 (URN)978-91-7155-782-7 (ISBN)
Public defence
2008-12-19, sal 14, hus 5, Kräftriket, Stockholm, 10:00
Available from: 2008-11-27 Created: 2008-11-18Bibliographically approved

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