Holomorphic Function Spaces on Homogeneous Siegel Domains

Abstract

We study several connected problems of holomorphic function spaces on homogeneous Siegel domains. The main object of our study concerns weighted mixed norm Bergman spaces on homogeneous Siegel domains of type II. These problems include: sampling, atomic decomposition, duality, boundary values, boundedness of the Bergman projectors. Our analysis include the Hardy spaces, and suitable generalizations of the classical Bloch and Dirichlet spaces. One of the main novelties in this work is the generality of the domains under consideration, that is, homogeneous Siegel domains, extending many results from the more particular cases of the upper half-plane, Siegel domains of tube type over irreducible cones, or symmetric, irreducible Siegel domains of type II.2010 Mathematical Subject Classification: 42B35, 32M15.

Toeplitz and Cesàro-Type Operators on Homogeneous Siegel Domains

“…We present here without proof some basic facts concerning homogeneous Siegel domains of type II. For a more detailed exposition, see [6], of which we keep the notation as far as possible.…”

“…In this section we define the main function spaces we shall consider, and state some of their properties. We refer the reader to [6] for a more thorough exposition. Definition 3.1.…”

“…See [6,Proposition 4.20 and Theorem 4.23] for a proof of the following result. Here and in what follows, we put p ′ := max(1, p) ′ for every p ∈]0, ∞], so that p ′ = ∞ if p 1, and…”

“…maps ℓ p,q (J, K) into A p,q s (D) continuously). 6 We say that property (L ′ ) p,q s,s ′ ,0 (resp. (L ′ ) p,q s,s ′ ) holds if, for every δ 0 > 0 as above, we may take (ζ j,k , z j,k ) j∈J,k∈K in such a way that the corresponding mapping Ψ is onto.…”

“…and admits a natural 2-step nilpotent Lie group structure whose product is best described under the identi- [6,Section 1.1]. We denote by N the set E × F endowed with this group structure.…”