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.Comment: Minor correctio

Martin boundary of a fine domain and a Fatou-Naim-Doob theorem for finely superharmonic functions

We construct the Martin compactification ${\bar U}$ of a fine domain $U$ in $R^n$, $n\ge 2$, and the Riesz-Martin kernel $K$ on $U \times{\bar U}$. We obtain the integral representation of finely superharmonic fonctions $\ge 0$ on $U$ in terms of $K$ and establish the Fatou-Naim-Doob theorem in this setting.Comment: Manuscript as accepted by publisher. To appear in Potential Analysi

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.Comment: 20 pages, manuscript as accepted by publisher. To appear in Potential Analysi

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.Comment: 28 page

A note on the structure of plurifinely open sets and the equality of some complex Monge-Amp\`ere 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 $U=\bigcup \{\varphi_j>-1\}$, where each $\phi_j$ 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-Amp\`ere measures. Unfortunately, this result is wrong as we will see below.Comment: New version with small change in the abstrac

Remarks on weak convergence of complex Monge-Amp\`ere measures

Let $(u_j)$ be a deaceasing sequence of psh functions in the domain of definition $\cal D$ of the Monge-Amp\`ere operator on a domain $\Omega$ of $\mathbb{C}^n$ such that $u=\inf_j u_j$ is plurisubharmonic on $\Omega$. In this paper we are interested in the problem of finding conditions insuring that \begin{equation*} \lim_{j\to +\infty} \int\varphi (dd^cu_j)^n=\int\varphi {\rm NP}(dd^cu)^n \end{equation*} for any continuous function on $\Omega$ 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 $K\subset\{u>-\infty\}$

An approach to control collaborative processes in PLM systems

Companies that collaborate within the product development processes need to implement an effective management of their collaborative activities. Despite the implementation of a PLM system, the collaborative activities are not efficient as it might be expected. This paper presents an analysis of the problems related to the collaborative work using a PLM system. From this analysis, we propose an approach for improving collaborative processes within a PLM system, based on monitoring indicators. This approach leads to identify and therefore to mitigate the brakes of the collaborative work