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Mohamed El Kadiri

Publications and source records attributed to Mohamed El Kadiri.

13 recordsLinked to original sources
arXiv2023

A note on equality on plurifinely open sets of some complex Monge-Ampère measures

Our aim in this paper is to prove that if plurisubharmonic functions $u_1,. . . , u_n$, $v_1,. . ., v_n$ in the domain of definition of the complex Monge-Ampère operator on a domain set $D\subset \mathbb{C}^n$ ($n\geq 1$) are such that $u_1= v_1, . . ., u_n=v_n$ on a Borel plurifinely open set $Ω\subset D$, then $$dd^cu_1\wedge ...\wedge dd^cu_n=dd^cv_1\wedge ...\wedge dd^cv_n$$ on $Ω$. This extends an earlier result obtained by the author in \cite{EK}.

math.CV↗
arXiv2023

A note on the structure of plurifinely open sets and the equality of some complex Monge-Ampère measures

In a recent preprint published on arXiv (see arXiv:2308.02993v2, referred here as \cite{NXH}), N.X. Hong stated that every plurifinely open set $U\subset \mathbb{C}^n$, $n\geq 1$, is of the form $U=\bigcup \{φ_j>-1\}$, where each $ϕ_j$ is a negative plurisubharmonic function defined on an open ball $B_j\subset \mathbb{C}^n$ and used this result to prove an equality result on complex Monge-Ampère measures. Unfortunately, this result is wrong as we will see below.

math.CV↗
arXiv2023

Remarks on weak convergence of complex Monge-Ampère measures

Let $(u_j)$ be a deaceasing sequence of psh functions in the domain of definition $\cal D$ of the Monge-Ampère operator on a domain $Ω$ of $\mathbb{C}^n$ such that $u=\inf_j u_j$ is plurisubharmonic on $Ω$. In this paper we are interested in the problem of finding conditions insuring that \begin{equation*} \lim_{j\to +\infty} \intφ(dd^cu_j)^n=\intφ{\rm NP}(dd^cu)^n \end{equation*} for any continuous function on $Ω$ with compact support, where ${\rm NP}(dd^cu)^n$ is the nonpolar part of $(dd^cu)^n$, and conditions implying that $u\in \cal D$. For $u_j=\max(u,-j)$ these conditions imply also that \begin{equation*} \lim_{j\to +\infty} \int_K(dd^cu_j)^n=\int_K {\rm NP}(dd^cu)^n \end{equation*} for any compact set $K\subset\{u>-\infty\}$.

math.CV↗
arXiv2022

An equality of Monge-Ampère measures

Let $u$ and $v$ be two plurisubharmonic functions in the domain of definition of the Monge-Ampère operator on a domain $Ω\subset {\bf C}^n$. We prove that if $u=v$ on a plurifinely open set $U\subset Ω$ that is Borel measurable, then $(dd^cu)^n|_U=(dd^cv)^n|_U$. This result was proved by Bedford and Taylor in the case where $u$ and $v$ are locally bounded, and by El Kadiri and Wiegerinck when $u$ and $v$ are finite, and by Hai and Hiep when $U$ is of the form $U=\bigcup_{j=1}^m\{φ_j>ψ_j\}$, where $φ_j$, $ψ_j$, $j=1,...,m$, are plurisubharmonic functions on $Ω$.

math.CV↗
arXiv2017

A note on maximal plurifinely plurisubharmonic functions

In this note we study the plurifinely locally maximal plurifinely plurisubharmonic functions and improve some known results on these functions. We prove in particular that any locally bounded plurifinely locally maximal plurifinely plurisubharmonic function is maximal, and that any finite plurisubharmonic function on an open subset of $\mathbb{C}^n$ is maximal if and only if it is plurifinely locally maximal.

math.CV↗
arXiv2017

On the existence of the biharmonic Green kernels and the the adjoint biharmonic functions

We study the existence and the regularity of the biharmonic Green kernel in a Brelot biharmonic space whose associated harmonic spaces have Green kernels. We show by some examples that this kernel does not always exist. We then introduce and study the adjoint of the given biharmonic space. This study was initiated by Smyrnelis, however, it seems that several results were incomplete and we clarify them here.

math.AP↗
arXiv2015

Sweeping at the Martin boundary of a fine domain

We study sweeping on a subset of the Riesz-Martin space of a fine domain in $\RR^n$ ($n\ge2$), both with respect to the natural topology and the minimal-fine topology, and show that the two notions of sweeping are identical.

math.AP↗
arXiv2014

Maximal Plurifinely Plurisubharmonic functions

The main purpose of this paper is to introduce and study the notion of plurifinely-maximal plurifinely plurisubharmonic functions, which extends the notion of maximal plurisubharmonic functions on a Euclidean domain to a plurifine domain of C^n in a natural way. Our main result is that a finite plurifinely plurisubharmonic function u on a plurifine domain U satisfies (dd^c u)^n=0 if and only if u is plurifinely-locally plurifinely-maximal outside some pluripolar set. In particular, a finite plurifinely-maximal plurisubharmonic function u satisfies (dd^c u)^n=0.

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↗