We give upper and lower bounds for constants appearing in the L²-estimates for the ∂̅-operator due to Donnelly-Fefferman and Berndtsson.

The characteristic function for a contraction is a classical complete unitary invariant devised by Sz.-Nagy and Foiaş. Just as a contraction is related to the Szegö kernel $k_S(z,w)={(1-zw̅)}^{-1}$ for |z|,|w| < 1, by means of $(1/k_S)(T,T*)\ge 0$, we consider an arbitrary open connected domain Ω in ℂⁿ, a complete Pick kernel k on Ω and a tuple T = (T₁, ..., Tₙ) of commuting bounded operators on a complex separable Hilbert space ℋ such that (1/k)(T,T*) ≥ 0. For a complete Pick kernel the 1/k functional calculus makes sense in a beautiful...

For some given logarithmically convex sequence M of positive numbers we construct a subspace of the space of rapidly decreasing infinitely differentiable functions on an unbounded closed convex set in ℝn. Due to the conditions on M each function of this space admits a holomorphic extension in ℂn. In the current article, the space of holomorphic extensions is considered and Paley-Wiener type theorems are established. To prove these theorems, some auxiliary results on extensions of holomorphic functions...

It is shown that on strongly pseudoconvex domains the Bergman projection maps a space $L{v}_k$ of functions growing near the boundary like some power of the Bergman distance from a fixed point into a space of functions which can be estimated by the consecutive power of the Bergman distance. This property has a local character. Let Ω be a bounded, pseudoconvex set with C³ boundary. We show that if the Bergman projection is continuous on a space $E\supset L^{\infty }\left(\Omega \right)$ defined by weighted-sup seminorms and equipped with the topology...

We define locally convex spaces LW and HW consisting of measurable and holomorphic functions in the unit ball, respectively, with the topology given by a family of weighted-sup seminorms. We prove that the Bergman projection is a continuous map from LW onto HW. These are the smallest spaces having this property. We investigate the topological and algebraic properties of HW.

Let K be a compact set in ℂ, f a function analytic in ℂ̅∖K vanishing at ∞. Let $f\left(z\right)={\sum }_{k=0}^{\infty }{a}_k{z}^{-k-1}$ be its Taylor expansion at ∞, and $H_s\left(f\right)=det{\left(a_{k+l}\right)}_{k,l=0}^s$ the sequence of Hankel determinants. The classical Pólya inequality says that $limsu{p}_{s\to \infty }{\left|H_s\left(f\right)\right|}^{1/s²}\le d\left(K\right)$, where d(K) is the transfinite diameter of K. Goluzin has shown that for some class of compacta this inequality is sharp. We provide here a sharpness result for the multivariate analog of Pólya’s inequality, considered by the second author in Math. USSR Sbornik 25 (1975), 350-364.

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...