Dans ce travail, nous étudions le problème de décomposicion suivant: Étant donnés deux ouverts bornés de Cp, Ω1 et Ω2 (vérifiant certaines conditions) et étant donnée une matrice A(z), carrée d'ordre n, dont les coefficients sont des fonctions holomorphes dans Ω1 ∩ Ω2, ayant une prolongement C∞ à l'adhérence (Ω1 ∩ Ω2), peut-on trouver deux matrices A1(z), A2(z) holomorphes dans Ω1 et Ω2 respectivement et se prolongeant de manière C∞ à (Ω1) et (Ω2) telles que sur Ω1 ∩ Ω2 on aitA = A1A2.

Let Ω be a domain of finite type in ℂ² and let f be a function holomorphic in Ω and belonging to ${}^{k,\alpha }\left(\Omega ̅\right)$. We prove the existence of boundary values for some suitable derivatives of f of order greater than k. The gain of derivatives holds in the complex-tangential direction and it is precisely related to the geometry of ∂Ω. Then we prove a property of non-isotropic Hölder regularity for these boundary values. This generalizes some results given by J. Bruna and J. M. Ortega for the unit ball.

For $z\in \partial B^n$, the boundary of the unit ball in ${ℂ}^n$, let $\Lambda \left(z\right)=\left\{\lambda \right|\lambda |\le 1\}$. If $f\in 𝕆\left(B^n\right)$ then we call $E\left(f\right)=\{z\in \partial B^n{\int }_{\Lambda \left(z\right)}|f\left(z\right){|}^2\mathrm{d}\Lambda \left(z\right)=\infty \}$ the exceptional set for $f$. In this note we give a tool for describing such sets. Moreover we prove that if $E$ is a $G_{\delta }$ and $F_{\sigma }$ subset of the projective $(n-1)$-dimensional space ${\mathbb{P}}^{n-1}=\mathbb{P}\left({ℂ}^n\right)$ then there exists a holomorphic function $f$ in the unit ball $B^n$ so that $E\left(f\right)=E$.

Let $D$ be a domain in ${ℂ}^2$. For $w\in ℂ$, let $D_w=\{z\in ℂ\mid (z,w)\in D\}$. If $f$ is a holomorphic and square-integrable function in $D$, then the set $E(D,f)$ of all $w$ such that $f(.,w)$ is not square-integrable in $D_w$ is of measure zero. We call this set the exceptional set for $f$. In this note we prove that for every $0<r<1$,and every $G_{\delta }$-subset $E$ of the circle $C(0,r)=\{z\in ℂ\mid |z|=r\}$,there exists a holomorphic square-integrable function $f$ in the unit ball $B$ in ${ℂ}^2$ such that $E(B,f)=E.$

Some representations of Nash functions on continua in ℂ as integrals of rational functions of two complex variables are presented. As a simple consequence we get close relations between Nash functions and diagonal series of rational functions.

We give representations of Nash functions in a neighbourhood of a polydisc (torus) in ${ℂ}^m$ as diagonal series of rational functions in a neighbourhood of a polydisc (torus) in ${ℂ}^{m+1}$.

In this paper, we limit our analysis to the difference of the weighted composition operators acting from the Hardy space to weighted-type space in the unit ball of ${ℂ}^N$, and give some necessary and sufficient conditions for their boundedness or compactness. The results generalize the corresponding results on the single weighted composition operators and on the differences of composition operators, for example, M. Lindström and E. Wolf: Essential norm of the difference of weighted composition operators....

Soient $G$ une variété de groupe définie sur le corps $\bar{\mathbf{Q}}$ des nombres algébriques, et $\phi :{\mathbf{C}}^n\to G_{\mathbf{C}}$ un sous-groupe à $n$ paramètres de $G$, de dimension algébrique $d$. Nous nous proposons de majorer le rang $\ell$ (sur $\mathbf{Z}$) des sous-groupes $\Gamma$ de ${\mathbf{C}}^n$ dont l’image par $\phi$ est contenue dans le groupe $G_{\bar{\mathbf{Q}}}$ des points algébriques de $G$.E. Bombieri et S. Lang ont déjà obtenu de telles majorations, en supposant que les points de $\Gamma$ sont très bien distribués : pour $d\ge n+1$, on a $\ell \le n^2+3n$ pour des variétés linéaires, et $\ell \le 2n^2+4n$ pour des variétés abéliennes .Nous...