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Said El Marzguioui

Publications and source records attributed to Said El Marzguioui.

5 recordsLinked to original sources
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

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 $\mathbb C$ we prove that there exists a pluripolar set $E \subset (S \times \mathbb C)$ with the property that the pluripolar hull of $E$ relative to $\mathbb C^2$ contains no fine analytic structure and its projection onto the first coordinate plane equals $\mathbb C$.

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↗