arXiv subjects

Fatima Zahra Assila

Publications and source records attributed to Fatima Zahra Assila.

3 recordsLinked to original sources

arXiv2025

The Range of the Monge-Ampère operator $(ω+ dd^c .)^n$ in bounded domains

Let $Ω$ be a bounded strictly pseudoconvex domain of $\mathbb{C}^n$. We solve degenerate complex Monge-Ampère equations of the form $(ω+ dd^c φ)^n = μ$ in the generalized Cegrell classes $\mathcal{K}(Ω,ω,H)$, where $H \in \mathcal{E}(Ω)$ is maximal, $ω$ is a smooth real $(1,1)$-form defined in a neighborhood of $\barΩ$ and $μ$ is a positive Radon measure. This generalizes the previous work of the last author \cite{Sal25} to the case of non-continuous functions $H$ and also to the case of measures $μ$ which do not vanish on pluripolar sets.

math.CV↗

arXiv2019

Some applications of Projective Logarithmic Potentials

We continue the study in \cite{As18, AAZ18} by giving a multitude of applications of projective logarithmic potentials. First we introduce the notions of projective logarithmic energy and capacity associated to projective kernel that was introduced and studied in \cite{As18, AAZ18}. We compare quantitatively the projective logarithmic capacity with the complex Monge-Ampère capacity on $\mathbb P^n$ and we deduce that the set of zero logarithmic capacity is of Monge-Ampère capacity zero. Further, we define transfinite diameter of a compact set and we show that it coincides with logarithmic capacity. Finally we deduce that there is an analogous of classical Evans's theorem that for any compact set $K$ of zero projective logarithmic capacity shows the existence of Probability measure whose potential admits $K$ as polar set.

math.CV↗

arXiv2017

Logarithmic potentials on $\mathbb{P}^n$

We study the projective logarithmic potential $\mathbb{G}_μ$ of a Probability measure $μ$ on the complex projective space $\mathbb{P}^{n}$. We prove that the Range of the operator $μ\longrightarrow \mathbb{G}_μ$ is contained in the (local) domain of definition of the complex Monge-Ampère operator acting on the class of quasi-plurisubharmonic functions on $\mathbb{P}^n$ with respect to the Fubini-Study metric. Moreover, when the measure $μ$ has no atom, we show that the complex Monge-Ampère measure of its Logarithmic potential is an absolutely continuous measure with respect to the Fubini-Study volume form on $\mathbb{P}^{n}$

math.CV↗