Nous prouvons en particulier que tout domaine homogène borné de ${\mathbf{C}}^n$, à frontière deux fois continûment différentiable est bi-holomorphiquement équivalent à la boule unité de ${\mathbf{C}}^n$. Les démonstrations sont entièrement élémentaires.

Dans cet article on montre que toute $f\in A^{\infty }(\bar{D})$ a une décomposition $f\left(z\right)-f\left(w\right)={\sum }_{i=1}^n{g}_i(z,w)(z_i-w_i)$ avec $g_i\in A^{\infty }(\overline{D\times D})$ pour les domaines pseudoconvexes à frontière réelle-analytique et aussi pour les domaines pseudoconvexes pour lesquels le résultat soit valable localement.

Dato un sistema omogeneo di equazioni di convoluzione in spazi dotati di strutture analiticamente uniformi, si forniscono condizioni per ottenere teoremi di rappresentazione per le sue soluzioni.

A Fréchet space with a sequence ${{\left|\right|·\left|\right|}_k}_{k=1}^{\infty }$ of generating seminorms is called tame if there exists an increasing function σ: ℕ → ℕ such that for every continuous linear operator T from into itself, there exist N₀ and C > 0 such that ${\left|\right|T\left(x\right)\left|\right|ₙ\le C\left|\right|x\left|\right|}_{\sigma \left(n\right)}$ ∀x ∈ , n ≥ N₀. This property does not depend upon the choice of the fundamental system of seminorms for and is a property of the Fréchet space . In this paper we investigate tameness in the Fréchet spaces (M) of analytic functions on Stein manifolds M equipped with the compact-open...

A necessary and sufficient condition for tautness of locally taut domains in a weakly Brody hyperbolic complex space is given. Moreover, some results of Kobayashi and Gaussier are deduced as corollaries.

We investigate the Bergman kernel function for the intersection of two complex ellipsoids $\{(z,w_1,w_2)\in {ℂ}^{n+2}:|z_1{|}^2+\cdots +|z_n{|}^2+|w_1{|}^q<1,|z_1{|}^2+\cdots +|z_n{|}^2+|w_2{|}^r<1\}.$ We also compute the kernel function for $\{(z_1,w_1,w_2)\in {ℂ}^3:|z_1{|}^{2/n}+|w_1{|}^q<1,|z_1{|}^{2/n}+|w_2{|}^r<1\}$ and show deflation type identity between these two domains. Moreover in the case that $q=r=2$ we express the Bergman kernel in terms of the Jacobi polynomials. The explicit formulas of the Bergman kernel function for these domains enables us to investigate whether the Bergman kernel has zeros or not. This kind of problem is called a Lu Qi-Keng problem.

We compute the Bergman kernel functions of the unbounded domains ${\Omega }_p=(z^{\text{'}},z)\in ℂ²:z>p\left(z^{\text{'}}\right)$, where $p\left(z^{\text{'}}\right)={\left|z^{\text{'}}\right|}^{\alpha }/\alpha$. It is also shown that these kernel functions have no zeros in ${\Omega }_p$. We use a method from harmonic analysis to reduce the computation of the 2-dimensional case to the problem of finding the kernel function of a weighted space of entire functions in one complex variable.