The Pluripolar Hull of a Graph and Fine Analytic Continuation
Abstract
We show that if the graph of a bounded analytic function in the unit disk $\mathbb D$ is not complete pluripolar in $\mathbb C^2$ then the projection of the closure of its pluripolar hull contains a fine neighborhood of a point $p \in \partial \mathbb D$. On the other hand we show that if an analytic function $f$ in $\mathbb D$ extends to a function $\mathcal{F}$ which is defined on a fine neighborhood of a point $p \in \partial \mathbb D$ and is finely analytic at $p$ then the pluripolar hull of the graph of $f$ contains the graph of $\mathcal{F}$ 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 nontrivial pluripolar hulls, among them $C^\infty$ functions on the closed unit disk which are nowhere analytically extendible and have infinitelysheeted pluripolar hulls. Previous examples of functions with nontrivial pluripolar hull of the graph have fine analytic continuation.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 May 2004
 arXiv:
 arXiv:math/0405025
 Bibcode:
 2004math......5025E
 Keywords:

 Complex Variables;
 32U15