Graphs that are not complete pluripolar

Abstract

Abstract. Let D 1 ⊂ D 2 be domains in C. Under very mild conditions on D 2 we show that there exist holomorphic functions f , defined on D 1 with the property that f is nowhere extendible across ∂D 1 , while the graph of f over D 1 is not complete pluripolar in D 2 × C. This refutes a conjecture of Levenberg, Martin and Poletsky (1992).

Determination of the pluripolar hull of graphs of certain holomorphic functions

“…In Section 4 we prove a localization principle for pluriharmonic measure. This turns out to be strong enough to explain qualitatively Siciak's [13] extension of our example in [2] of a holomorphic function f ∈ A ∞ (D) with domain of existence the unit disc D, which has (Γ f ) * extending over most of C. We also show that the pluripolar hull of a connected F σ -pluripolar set is connected; this may be of independent interest.…”

“…In [2], the authors constructed an example of a smooth holomorphic function f on the unit disc such that (Γ f ) * C 2 \ Γ f = ∅. From Proposition 4.4 (see the discussion after the Proposition) and Corollary 4.8 we see that the set (Γ f ) * C 2 \ Γ f is actually quite big.…”

“…Let f be a holomorphic function on its domain of existence D ⊂ C and let Γ f be its graph in D × C. Answering a question of Levenberg, Martin and Poletsky [6], we showed in [2] that it is possible that Γ f is not a complete pluripolar subset of C 2 , but that the pluripolar hull of Γ f is strictly larger than Γ f . In a subsequent paper [3] we studied the pluripolar hull (Γ f ) * D 0 relative to a domain D 0 in the following setup: D ⊂ D 0 are domains in C, K = D 0 \ D is a closed polar subset of D, and z ∈ K. We showed that a necessary and sufficient conditions for {z} × C ∩ (Γ f ) * D 0 ×C = ∅ is that z be a regular boundary point for the Dirichlet problem on…”

“…[4]. In [2,3] we gave a sufficient condition for graphs of holomorphic functions to have a non-trivial pluripolar hull; Theorem 5.3 provides the natural generalization to pluripolar sets.…”

Pluripolarity of graphs of Denjoy quasianalytic functions of several variables

“…which is a required contradiction. Therefore, the function f satisfies condition (14) and, consequently, it also satisfies condition (3). Thus, any Denjoy quasianalytic function satisfies the conditions of the proposition.…”