The Pluripolar Hull of a Graph and Fine Analytic Continuation

We show that if the graph of a bounded analytic function in the unit disk D is not complete pluripolar in C then the projection of the closure of its pluripolar hull contains a fine neighborhood of a point p ∈ ∂D. 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 ∈ ∂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 nontrivial 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.