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Noureddine Ghiloufi

Publications and source records attributed to Noureddine Ghiloufi.

15 recordsLinked to original sources
arXiv2026

Slices and $m$-Lelong numbers of $m$-subharmonic functions

We investigate slicing properties of $m$-subharmonic functions in product domains $Ω= Ω' \times Ω'' \subset \mathbb{C}^n = \mathbb{C}^p \times \mathbb{C}^{n-p}$, where $p, m, n$ are integers satisfying $1 \leq p \leq m-1 < n-1$.\\ Given an $m$-subharmonic function $v$ on $Ω$, we prove the existence of a pluripolar subset $E \subset Ω'$ such that, for every $x' \in Ω' \smallsetminus E$, the slice $v_{|\{x'\}\times \mathbb{C}^{n-p}}$ is well defined and $(m - q_{m,p})$-subharmonic on $Ω''$, where $q_{m,p}$ denotes the smallest integer greater than or equal to $\frac{mp}{n}$.\\ Moreover, we show that, outside a negligible subset of $Ω'$, the $m$-Lelong number of $v$ at $(x', x'')$ coincides, up to a multiplicative constant, with the $(m - q_{m,p})$-Lelong number of the slice $v_{|\{x'\}\times Ω''}$ at $x''$.

math.CV↗
arXiv2026

Spectral properties of commutators on harmonic Bergman spaces of the unit disk

In this paper, we determine the asymptotic behavior of the singular values of the commutator $\mathscr{C}_u := [M_u, \mathbb{P}_α]$ acting on $L^2(\mathbb D, dA_α)$, where $M_u$ is the operator of multiplication by a subharmonic function $u$ on $\mathbb D(R)$ and harmonic outside the origin, with $R>1$, and $\mathbb{P}_α$ is the orthogonal projection onto the space $\mathscr{H}_α^2(\mathbb D)$ of harmonic functions on $\mathbb D$ that are square-integrable with respect to the weighted measure $dA_α$. We prove that if $u(z) = U(z) + \overline{U(z)} + ν_u \log|z|^2$, where $U$ is a holomorphic function on $\mathbb D(R)$ and $2ν_u$ is the Lelong number of $u$ at $0$, then $$s_n(\mathscr{C}_u) \underset{n\to+\infty}{\sim} \frac{\sqrt{α+1}}{2πn} \int_{\partial \mathbb D} \sqrt{ν_u^2 + |U'(z)|^2} \; |dz|.$$ In particular, the operator $\mathscr{C}_u$ belongs to the Von Neumann-Schatten class $\mathcal C_p$ for any $p>1$.

math.CV↗
arXiv2025

Spectral properties of the Cauchy transform on modified Bergman spaces

In this paper, we determine the singular values $s_n(T_{α,β})$ and $s_n(R_{α,β})$ of the operators $T_{α,β}=\mathcal C\mathbb P_{α,β}$ and $R_{α,β}=\mathbb P_{α,β}\mathcal C\mathbb P_{α,β}$ where $\mathcal C$ is the integral Cauchy transform and $\mathbb P_{α,β}$ is the orthogonal projection from $L^2(\mathbb D,μ_{α,β})$ onto the modified Bergman space $\mathcal A^2(\mathbb D,μ_{α,β})$. These singular values will be expressed in terms of some series involving hypergeometric functions. We show that in both cases the sequence $n^{α+1}s_n(.)$ has a finite limit as $n\to+\infty$.

math.CV↗
arXiv2024

Modified Bergman spaces on the unit ball of $\mathbb C^n$ and applications

In this paper, we introduce new spaces of holomorphic functions on the unit ball $\mathbb{B}_{n}$ of $\mathbb{C}^{n}$ generalizing the classical Bergman spaces. The main results include the properties of some operators and integrals representations such as Bergman-type projections, and Berezin transform.

math.CV↗
arXiv2023

Meromorphic Bergman spaces

In this paper we introduce new spaces of holomorphic functions on the pointed unit disc of $\mathbb C$ that generalize classical Bergman spaces. We prove some fundamental properties of these spaces and their dual spaces. We finish the paper by extending Hardy-Littlewood and Fejér-Riesz inequalities to these spaces with an application on Toeplitz operators.

math.CV↗
arXiv2023

Asymptotic results on modified Bergman-Dirichlet spaces and examples of Segal-Bargmann transforms

In this paper, we start by introducing the modified Bergman-Dirichlet space $\mathcal D_m^2(\mathbb D_R,μ^R_{α,β})$ and then we study its asymptotic behavior when the parameter $α$ goes to infinity and to $(-1)$ to obtain respectively the modified Bargmann-Dirichlet and the modified Hardy-Dirichlet spaces with their reproducing kernels. Finally, we give some examples of Segal-Bargmann transforms of those spaces.

math.CV↗
arXiv2020

Zeros of new Bergman kernels

In this paper we determine explicitly the kernels $\mathbb K_{α,β}$ associated with new Bergman spaces $\mathcal A_{α,β}^2(\mathbb D)$ considered recently by the first author and M. Zaway. Then we study the distribution of the zeros of these kernels essentially when $α\in\mathbb N$ where the zeros are given by the zeros of a real polynomial $Q_{α,β}$. Some numerical results are given throughout the paper.

math.CV↗
arXiv2017

Lelong numbers of $m-$subharmonic functions

In this paper we study the existence of Lelong numbers of $m-$subharmonic currents of bidimension $(p,p)$ on an open subset of $\Bbb C^n$, when $m+p\geq n$. In the special case of $m-$subharmonic function $φ$, we give a relationship between the Lelong numbers of $dd^cφ$ and the mean values of $φ$ on spheres or balls. As an application we study the integrability exponent of $φ$. We express the integrability exponent of $φ$ in terms of volume of sub-level sets of $φ$ and we give a link between this exponent and its Lelong number.

math.CV↗
arXiv2014

Pluricomplex energy classes associated to a positive closed current

The aim of this paper is to extend the domain of definition of $(dd^c\centerdot)^q\wedge T$ on some classes of plurisubharmonic (psh) functions, which are not necessary bounded, where $T$ is a positive closed current of bidimension $(q,q)$ on an open set $Ω$ of $\Bbb C^n$. We introduce two classes $\mathcal{F}_{p}^{T}(Ω)$ and $\mathcal{E}_p^T(Ω)$ and we show that they belong to the domain of definition of the operator $(dd^c\centerdot)^q\wedge T$. We also prove that all functions belong to these classes are $C_T$-quasicontinuous and that the comparison principle is valid in them.

math.CV↗
arXiv2014

On the directional Lelong-Demailly numbers of positive currents

In this paper we study the existence of the directional Lelong-Demailly numbers of positive plurisubharmonic or plurisuperharmonic currents. We prove the independence of these numbers to the system of coordinates. Moreover these numbers will be given by locally integrable functions.

math.CV↗
arXiv2012

Existence of the directional tangent cone to a positive current

In this paper, we start by proving the existence of the strict transform of a positive current $T$ as soon as its $j^{th}$ currents, $T_j$, are plurisubharmonics or plurisuperharmonics. Then, with a suitable condition on $T_j$, we show the existence of the directional tangent cone to $T$. In the particular case, when $T$ is closed, we give a second condition independent to the previous one.

math.CV↗
arXiv2011

On the tangent cones to plurisubharmonic currents

In this paper, we study the existence of the tangent cone to a positive plurisubharmonic or plurisuperharmonic current with a suitable condition. Some Estimates of the growth of the Lelong functions associated to the current and to its $dd^c$ are given to ensure the existence of the blow-up of this current. A second proof for the existence of the tangent cone is derived from these estimates.

math.CV↗
arXiv2011

On the Lelong-Demailly numbers of plurisubharmonic currents

In this note we study the existence of the Lelong-Demailly number of a negative plurisubharmonic current with respect to a positive plurisubharmonic function on an open subset of $\C^n$. Then we establish some estimates of the Lelong-Demailly numbers of positive or negative plurisubharmonic currents.

math.CV↗
arXiv2011

Ordres des courants positifs pluriharmoniques

In this article, we study the order of a positive pluriharmonic current and we compare it with either the order of the concurrent slices or the directionnel orders of the current. Therefore some estimates of the growth of the \textsc{Lelong} function are established and the problem of algebraicity of the current is treated as a result.

math.CV↗