The pluripolar hull of a graph and fine analytic continuation |
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Authors: | Tomas Edlund Burglind Jöricke |
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Institution: | (1) Department of Mathematics, Uppsala University, P.O. Box 480, SE-751 06 Uppsala, Sweden |
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Abstract: | We show that if the graph of an analytic function in the unit disk D is not complete pluripolar in C2 then the projection of its pluripolar hull contains a fine neighborhood of a point
. 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 ℱ which is defined on a fine neighborhood of a point
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∞ 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. |
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