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#### TBDmathscidoc:1701.333075

Arkiv for Matematik, 44, (1), 39-60, 2004.5
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 $p\in\partial\mathbf{D}$ . 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 ℱ which is defined on a fine neighborhood of a point $p\in\partial\mathbf{D}$ 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$C$^{∞}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.
@inproceedings{tomas2004the,
title={The pluripolar hull of a graph and fine analytic continuation},
author={Tomas Edlund, and Burglind Jöricke},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203616457272884},
booktitle={Arkiv for Matematik},
volume={44},
number={1},
pages={39-60},
year={2004},
}

Tomas Edlund, and Burglind Jöricke. The pluripolar hull of a graph and fine analytic continuation. 2004. Vol. 44. In Arkiv for Matematik. pp.39-60. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203616457272884.