An example concerning Sadullaev's boundary relative extremal functions

We exhibit a smoothly bounded domain $\Omega$ with the property that for suitable $K\subset\partial \Omega$ and $z\in \Omega$ the "Sadullaev boundary relative extremal functions" satisfy the inequality $\omega_1(z,K,\Omega)<\omega_2(z,K,\Omega)\le \omega(z,K,\Omega)$.Comment: 3 page

Plurifine Potential Theory

An overview of the recent developments in plurifine potential theory.Comment: 18 pages, Conference on SCV on the occasion of Jozef Siciak's 80'th birthda

The pluripolar hull of the graph of a holomorphic function with polar singularities

We study the pluripolar hull of the graph of a holomorphic function f, defined on a domain D in the complex plane outside a polar set A of D. This leads to a theorem that describes under what conditions f is nowhere extendable over A, while the graph of f over D \ A is NOT complete pluripolar.Comment: 15 page

Determination of the pluripolar hull of graphs of certain holomorphic functions

Let A be a closed polar subset of a domain D in the complex plane C. We give a complete description of the pluripolar hull in D X C of the graph of a holomorphic function defined on D A. To achieve this, we prove for pluriharmonic measure certain semi-continuity properties and a localization principle.Comment: 13 page

Graphs that are not complete pluripolar

Let D_1 be a subdomain of D_2 in the complex plane CC. 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 the boundary of D_1, while the graph of f over D_1 is NOT complete pluripolar in D_2 times CC. This refutes a conjecture of Levenberg, Martin and Poletsky.Comment: 7 page

Graphs with multiple sheeted pluripolar hulls

In this paper we study the pluripolar hulls of analytic sets. In particular, we show that hulls of graphs of analytic functions can be multiple sheeted and sheets can be separated by a set of zero dimension.Comment: 12 page

Characterizations of boundary pluripolar Hulls

We present some basic properties of the boundary relative extremal function and discuss so called 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.Comment: 9 pages, In this version some small changes were made and a few typo's were corrected. Version 3 has 10 pages. Section 2 has been rewritten. It now includes observations about different versions of the boundary extremal function as introduced by Sadullaev. We removed superfluous assumptions in some of the statement