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.

In this note we compute the Bergman kernel of the unit ball with respect to the smallest norm in ${ℂ}^n$ that extends the euclidean norm in ${ℝ}^n$ and give some applications.

We establish $L^p$-estimates for the weighted Bergman projection on a nonsingular cone. We apply these results to the weighted Fock space with respect to the minimal norm in ℂⁿ.

We introduce the Bloch space for the minimal ball and we prove that this space can be identified with the dual of a certain analytic space which is strongly related to the Bergman theory on the minimal ball.