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The pluripolar hull of a graph and fine analytic continuation
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
2006 (English)In: Arkiv för matematik, ISSN 0004-2080, E-ISSN 1871-2487, Vol. 44, no 1, 39-60 p.Article in journal (Refereed) Published
Abstract [en]

We show that if the graph of an analytic function in the unit disk D is not complete pluripolar in C-2 then the projection of its pluripolar hull contains a fine neighborhood of a point p is an element of partial derivative D. Moreover the projection of the pluripolar hull is always finely open. On the other hand we show that if an analytic function f in D extends to a function T which is defined on a fine neighborhood of a point p is an element of partial derivative D and is finely analytic at p then the pluripolar hull of the graph of f contains the graph of F over a smaller fine neighborhood of p. We give several examples of functions with this property of fine analytic continuation. As a corollary we obtain new classes of analytic functions in the disk which have non-trivial pluripolar hulls, among them C-infinity functions on the closed unit disk which are nowhere analytically extendible and have infinitely-sheeted pluripolar hulls. Previous examples of functions with non-trivial pluripolar hull of the graph have fine analytic continuation.

Place, publisher, year, edition, pages
2006. Vol. 44, no 1, 39-60 p.
National Category
Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-93258DOI: 10.1007/s11512-005-0004-3ISI: 000240662300003OAI: oai:DiVA.org:uu-93258DiVA: diva2:166687
Available from: 2005-09-01 Created: 2005-09-01 Last updated: 2011-06-15Bibliographically approved
In thesis
1. Pluripolar Sets and Pluripolar Hulls
Open this publication in new window or tab >>Pluripolar Sets and Pluripolar Hulls
2005 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

For many questions of complex analysis of several variables classical potential theory does not provide suitable tools and is replaced by pluripotential theory. The latter got many important applications within complex analysis and related fields. Pluripolar sets play a special role in pluripotential theory. These are the exceptional sets this theory. Complete pluripolar sets are especially important. In the thesis we study complete pluripolar sets and pluripolar hulls. We show that in some sense there are many complete pluripolar sets. We show that on each closed subset of the complex plane there is continuous function whose graph is complete pluripolar. On the other hand we study the propagation of pluripolar sets, equivalently we study pluripolar hulls. We relate the pluripolar hull of a graph to fine analytic continuation of the function. Fine analytic continuation of an analytic function over the unit disk is related to the fine topology introduced by Cartan and to the previously known notion of finely analytic functions. We show that fine analytic continuation implies non-triviality of the pluripolar hull. Concerning the inverse direction, we show that the projection of the pluripolar hull is finely open. The difficulty to judge from non-triviality of the pluripolar hull about fine analytic continuation lies in possible multi-sheetedness. If however the pluripolar hull contains the graph of a smooth extension of the function over a fine neighborhood of a boundary point we indeed obtain fine analytic continuation.

Place, publisher, year, edition, pages
Uppsala: Matematiska institutionen, 2005. 23 p.
Series
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 41
Keyword
Mathematical analysis, plurisubharmonic functions, pluripolar set, complete pluripolar set, pluripolar hull, fine topology, finely analytic function, fine analytic continuation, Matematisk analys
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:uu:diva-5872 (URN)91-506-1812-1 (ISBN)
Public defence
2005-09-22, 2347, Polacksbacken, Uppsala, 13:15
Opponent
Supervisors
Available from: 2005-09-01 Created: 2005-09-01Bibliographically approved

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