相关论文: Pluripolar graphs are holomorphic

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.

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…

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…

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…

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…

In this paper has been proved the pluripolarity of graphs of algebroid functions

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…

We give a simplified proof of analyticity of pluripolar multifunctions

We study functions defined on a closed segment of the real line that belong to the class of Gonchar. We show that the graphs of such functions are pluripolar. We also discuss the generalizations of our result to functions defined on a…

We show that the graph $$\Gamma_f=\{(z,f(z))\in{\Bbb C}^2: z\in S\}$$ in ${\Bbb C}^2$ of a function $f$ on the unit circle $S$ which is either continuous and quasianalytic in the sense of Bernstein or $C^\infty$ and quasianalytic in the…

For a domain $D\subset {\Bbb{C}}^n$ we construct a continuous foliation of $D$ into one real dimensional curves such that any function $f\in {C^1(D)}$ which can be extended holomorphically into some neighborhood of each curve in the…

We show that if the graph of a bounded analytic function in the unit disk $\mathbb D$ is not complete pluripolar in $\mathbb C^2$ then the projection of the closure of its pluripolar hull contains a fine neighborhood of a point $p \in…

Let C be real-analytic simple closed curve in the complex plane which is symmetric with respect to the real axis. Let r>0 be such that C+ir misses C-ir. We prove that if a continuous function f extends holomorphically from C+it for each t…

$J$-holomorphic curves are pluripolar, but they are not minus-infinity sets of pluri-subharmonic functions with logarithmic singularity.

Let $B^n$ be the $n$-dimensional unit complex ball and let $a$ and $b$ be two distinct points in its closure. Let $f$ be a real-analytic function on the complex unit sphere $\partial B^n.$ Suppose that for any complex line $L,$ meeting the…

It is known that if f is a continuous function on the complex plane which extends holomorphically from each circle surrounding the origin then f is not necessarily holomorphic. In the paper we prove that if, in addition, f extends…

Suppose that $F$ is a smooth and connected complex surface (not necessarily compact) containing a smooth rational curve with positive self-intersection. We prove that if there exists a non-constant meromorphic function on $F$, then the…

It is a well-known and elementary fact that a holomorphic function on a compact complex manifold without boundary is necessarily constant. The purpose of the present article is to investigate whether, or to what extent, a similar property…

Let D be a bounded, finitely connected domain in the complex plane without isolated points in the boundary and let f be a continuous function on the boundary bD. Let F be a continuous extension of f to the closure of D. We prove that f…

Let $X$ be a Riemann domain over $\mathbb C^k\times\mathbb C^\ell$. If $X$ is domain of holomorphy with respect to a family $\mathcal F\subset\mathcal O(X)$, then there exists a pluripolar set $P\subset\mathcal C^k$ such that every slice…