Pluripolar Hulls Research Articles

Pluripolar hulls and fine holomorphy

Examples by Poletsky and the author and by Zwonek show the existence of nowhere extendable holomorphic functions with the property that the pluripolar hull of their graphs is much larger than the graph of the respective functions, and contains multiple sheets. We will explain this phenomenon by fine analytic continuation of the function over part of a Cantor-type set involved. This gives more information on the hull, and allows for weakening and effectiveness of the conditions in the original examples.

Pluripolar Hulls and Convergence Sets

The pluripolar hull of a pluripolar set E in ℙn is the intersection of all complete pluripolar sets in ℙn that contain E. We prove that the pluripolar hull of each compact pluripolar set in ℙn is Fσ. The convergence set of a divergent formal power series f(z0, … , zn) is the set of all “directions” ξ ∈ ℙn along which f is convergent. We prove that the union of the pluripolar hulls of a countable collection of compact pluripolar sets in ℙn is the convergence set of some divergent series f. The convergence sets on Γ := {[1 : z : ψ (z)] : z ∈ ℂ} ⊂ ℂ2 ⊂ ℙ2, where ψ is a transcendental entire holomorphic function, are also studied and we obtain that a subset on Γ is a convergence set in ℙ2 if and only if it is a countable union of compact projectively convex sets, and hence the union of a countable collection of convergence sets on Γ is a convergence set.

Characterizations of boundary pluripolar hulls

We present some basic properties of the so-called boundary relative extremal function and discuss boundary pluripolar sets and boundary pluripolar hulls. We show that for B-regular domains the boundary pluripolar hull is always trivial on the boundary of the domain and present a “boundary version” of Zeriahi’s theorem on the completeness of pluripolar sets.

The image of a finely holomorphic map is pluripolar

We prove that the image of a finely holomorphic map on a fine domain in C is a pluripolar subset of C-n. We also discuss the relationship between pluripolar hulls and finely holomorphic functions.

Pluripolar Hulls and Complete Pluripolar Sets

We study the pluripolar hull of a complex subvariety in the complement of a closed complete pluripolar set. A result on propagation of pluripolar hulls is also given.

Pluripolar hulls and fine analytic structure

We discuss the relation between pluripolar hulls and fine analytic structure. Our main result is the following. For each non polar subset S of the complex plane ℂ we prove that there exists a pluripolar set E ⊂ ( S × ℂ) with the property that the pluripolar hull of E relative to ℂ 2 contains no fine analytic structure and its projection onto the first coordinate plane equals ℂ.

The pluripolar hull of a graph and fine analytic continuation

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 \(p\in\partial\mathbf{D}\). 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.

Bounded holomorphic functions with multiple sheeted pluripolar hulls

Bounded holomorphic functions with multiple sheeted pluripolar hulls

A Note on Pluripolar Hulls of Graphs of Blaschke Products

In the paper we study the problem of characterization of pluripolar hulls of a class of functions containing Blaschke products.

Remarks on pluripolar hulls

The aim of the paper is to establish some results on pluripolar hulls and to define pluripolar hulls of certain graphs.

Pluripolar hulls.

Pluripolar hulls.