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 $K\subset\{u>-\infty\}$

La fréquence des symptômes physiques dans les troubles anxio-dépressifs: étude transversale chez une population de 202 consultants psychiatriques

Introduction: les symptômes physiques associés aux troubles anxio-dépressifs ont fait l'objet de plusieurs études depuis plusieurs décennies, vu leurs fréquences et leurs conséquences. Le but de notre étude est de préciser la fréquence des principaux symptômes physiques des troubles anxieux: trouble panique (TP), trouble anxiété généralisée (TAG), troubles phobiques (TPh), et dépressifs: épisode dépressif majeur (EDM) dans le cadre d'un trouble dépressif). Méthodes: nous avons mené une étude transversale à visée descriptive, réalisée sur un échantillon de 202 consultants dans un Service de Psychiatrie. Résultats: l'âge moyen des patients est de 42 ans (19 à 70 ans), avec une légère prédominance féminine: 118 (58%). Les troubles anxio-dépressifs constatés sont: l'EDM: 113(56%), le TP: 61 (30.2%), le TAG: 55 (27.2%) et les TPh: 30 (14.9%). La fréquence des patients présentant de 2 à 5, et plus de 5 symptômes était respectivement de 15.9% et 39.6% dans les troubles dépressifs, et de 9.5% et 62.9% dans les troubles anxieux. Les symptômes les plus rapportés sont d'ordre cardiopulmonaire (75%), général (73.8%) et neurologique (65.8%). Conclusion: les symptômes physiques qui accompagnent les troubles anxio-dépressifs, sont variables et souvent nombreux. Ils peuvent aggraver le pronostic de ces troubles psychiatriques, en rendant difficile leur prise en charge. Un dépistage précoce de ces troubles, en portant une attention particulière à ces symptômes physiques, permettra de prévenir ces complications