Searcharxiv⌕ Search

arXiv subjects

Jan Wiegerinck

Publications and source records attributed to Jan Wiegerinck.

At least 19 recordsLinked to original sources
arXiv2022

Pluripolar Hulls and Fine Holomorphy

Examples by Poletsky and the author and by Zwonek show the existence 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.

math.CV↗
arXiv2018

Domains of existence for finely holomorphic functions

We show that fine domains in $\mathbf{C}$ with the property that they are Euclidean $F_σ$ and $G_δ$, are in fact fine domains of existence for finely holomorphic functions. Moreover \emph{regular} fine domains are also fine domains of existence. Next we show that fine domains such as $\mathbf{C}\setminus \mathbf{Q}$ or $\mathbf{C}\setminus (\mathbf{Q}\times i\mathbf{Q})$, more specifically fine domains $V$ with the properties that their complement contains a non-empty polar set $E$ that is of the first Baire category in its Euclidean closure $K$ and that $(K\setminus E)\subset V$, are NOT fine domains of existence.

math.CV↗
arXiv2016

A note on approximation of plurisubharmonic functions

We extend a recent result of Avelin, Hed, and Persson about approximation of functions $u$ that are plurisubharmonic on a domain $Ω$ and continuous on $\barΩ$, with functions that are plurisubharmonic on (shrinking) neighborhoods of $\barΩ$. We show that such approximation is possible if the boundary of $Ω$ is $C^0$ outside a countable exceptional set $E\subset\partial Ω$. In particular, approximation is possible on the Hartogs triangle. For Hölder continuous $u$, approximation is possible under less restrictive conditions on $E$. We next give examples of domains where this kind of approximation is not possible, even when approximation in the Hölder continuous case is possible.

math.CV↗
arXiv2015

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.

math.CV↗
arXiv2013

Plurifinely Plurisubharmonic functions and the Monge Ampère Operator

We will define the Monge-Ampère operator on finite (weakly) plurifinely plurisubharmonic functions in plurifinely open sets in complex n-space and show that it defines a positive measure. Ingredients of the proof include a direct proof for bounded strongly plurifinely plurisubharmonic functions, which is based on the fact that such functions can plurifinely locally be written as difference of ordinary plurisubharmonic functions, and an approximation result stating that weakly plurifinely plurisubharmonic functions are locally limits of strongly finely plurisubharmonic functions in the Dirichlet norm.

math.CV↗
arXiv2010

Plurisubharmonic and holomorphic functions relative to the plurifine topology

A weak and a strong concept of plurifinely plurisubharmonic and plurifinely holomorphic functions are introduced. Strong will imply weak. The weak concept is studied further. A function f is weakly plurifinely plurisubharmonic if and only if f o h is finely subharmonic for all complex affine-linear maps h. As a consequence, the regularization in the plurifine topology of a pointwise supremum of such functions is weakly plurifinely plurisubharmonic, and it differs from the pointwise supremum at most on a pluripolar set. Weak plurifine plurisubharmonicity and weak plurifine holomorphy are preserved under composition with weakly plurifinely holomorphic maps.

math.CV↗
arXiv2009

Continuity Properties of Finely Plurisubharmonic Functions and pluripolarity

We prove that every bounded finely plurisubharmonic function can be locally (in the pluri-fine topology) written as the difference of two usual plurisubharmonic functions. As a consequence finely plurisubharmonic functions are continuous with respect to the pluri-fine topology. Moreover we show that -infinity sets of finely plurisubharmonic functions are pluripolar, hence graphs of finely holomorphic functions are pluripolar.

math.CV↗
arXiv2008

Shcherbina's Theorem for Finely Holomorphic Functions

We prove an analogue of Sadullaev's theorem concerning the size of the set where a maximal totally real manifold can meet a pluripolar set. The manifold has to be of class C-1 only. This readily leads to a version of Shcherbina's theorem for C-1 functions f that are defined in a neighborhood of certain compact sets K in the complex plane. If the graph of f on K is pluripolar, then f satisfies the Cauchy Riemann equations in the closure of the fine interior of K.

math.CV↗
arXiv2008

Connectedness in the Pluri-fine Topology

We study connectedness in the pluri-fine topology on $\CC^n$ and obtain the following results. If $Ω$ is a pluri-finely open and pluri-finely connected set in $\CC^n$ and $E\subset\CC^n$ is pluripolar, then $Ω\setminus E$ is pluri-finely connected. The proof hinges on precise information about the structure of open sets in the pluri-fine topology: Let $Ω$ be a pluri-finely open subset of $\CC^{n}$. If $z$ is any point in $Ω$, and $L$ is a complex line passing through $z$, then obviously $Ω\cap L$ is a finely open neighborhood of $z$ in $L$. Now let $C_L$ denote the finely connected component of $z$ in $Ω\cap L$. Then $\cup_{L\ni z} C_L$ is a pluri-finely connected neighborhood of $z$. As a consequence we find that if $v$ is a finely plurisubharmonic function defined on a pluri-finely connected pluri-finely open set, then $v= -\infty$ on a pluri-finely open subset implies $v\equiv -\infty$.

math.CV↗
arXiv2008

The image of a finely holomorphic map is pluripolar

We prove that the image of a finely holomorphic map on a fine domain in $\mathbb{C}$ is pluripolar subset of $\mathbb{C}^{n}$. We also discuss the relationship between pluripolar hulls and finely holomorphic function.

math.CV↗
arXiv2005

More Approximation on Disks

In this paper we study the function algebra generated by z^2 and g^2 on a small closed disk centered at the origin of the complex plane. We prove, using a biholomorphic change of coordinates and already developed techniques in this area, that for a large class of functions g this algebra consists of all continuous functions on the disk.

math.CV↗
arXiv2005

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.

math.CV↗