The authors study holomorphic functions on homogeneous Siegel domains. They concentrate mainly on weighted mixed norm Bergman spaces. The problems considered include: sampling, atomic decomposition, duality, boundary values, and boundedness of the Bergman projectors. The work consists of an introduction and five chapters, it also contains an appendix devoted to mixed norm spaces. We shall briefly describe the contents of the consecutive chapters.
In the introduction, the authors motivate their study by discussing the simplest example of a Siegel domain: the upper half-plane \(\mathbb{C}_{+}:=\mathbb{R}+i\mathbb{R}_{+}^{*}\). They also present known results concerning function theory on Siegel domains.
Let \(E\) be a vector space over \(\mathbb{C}\) of finite dimension \(n\) and \(F\) be a vector space over \(\mathbb{R}\) of finite dimension \(m>0\). Let \(\Omega\) be an open convex cone with vertex \(0\) in \(F\) and not containing any affine lines. Also, let \(\Phi\colon E\times E\rightarrow F_{\mathbb{C}}\), \(F_{\mathbb{C}}\) is the complexification of \(F\), i.e., \(F_{\mathbb{C}}=F\otimes_{\mathbb{R}}\mathbb{C}\), be a positive non-degenerate hermitian mapping, that is:

(i)

\(\Phi\) is linear in the first argument;

(ii)

\(\Phi(\zeta,\zeta^{'})=\overline{\Phi(\zeta^{'},\zeta)}\) for all \(\zeta,\zeta^{'}\in E\);

(iii)

\(\Phi\) is non-degenerate;

(iv)

\(\Phi(\zeta):=\Phi(\zeta,\zeta)\in \overline{\Omega}\).

{\em The Siegel domain of type II} associated with the cone \(\Omega\) and the mapping \(\Phi\) is defined the following way: \[ D:=\{(\zeta,z)\in E\times F_{\mathbb{C}}\colon \operatorname{Im} z-\Phi(\zeta)\in \Omega\}. \] Chapter one contains, beside the above definition and basic examples, definitions of the Fourier transform, the Bergman and the Hardy spaces, the formulation of the corresponding Paley-Wiener theorems and a discussion of the Kohn Laplacian.
In Chapter two the authors introduce various objects related to homogeneous Siegel domains of type II. This includes a discussion of \(T\)-algebras and the associated homogeneous cones, the generalized power functions \(\Delta_{\Omega}^{s}\), \(\Delta_{\Omega^{'}}^{s}\), \(\Omega^{'}\) the dual cone and the associated gamma and beta functions. They also introduce the corresponding Bergman metric.
Chapter three is devoted to the study of the weighted Bergman spaces \(A_{s}^{p,q}(D)\). These spaces are defined for \(\mathbf{s}\in\mathbb{R}^{r}\) and \(p,q\in (0,\infty]\) as \begin{align*} A_{\mathbf{s}}^{p,q}(D)&:=\mathrm{Hol}(D)\cap L_{\mathbf{s}}^{p,q}(D)\\ A_{\mathbf{s},0}^{p,q}(D)&:=\mathrm{Hol}(D)\cap L_{\mathbf{s},0}^{p,q}(D), \end{align*} where \(L_{\mathbf{s}}^{p,q}(D)\) is the Hausdorff space associated with the space \[ \{f\colon D\rightarrow \mathbb{C}\colon f \text{\:measurable\:} \int_{\Omega}(\Delta_{\Omega}^{\mathbf{s}}\|f_{h}\|_{L^{p}(\mathcal{N}})^{q}d\nu_{\Omega}(h)<\infty\} \] and \(L_{\mathbf{s},0}^{p,q}(D)\) is the closure of \(C_{c}(D)\) in \(L_{\mathbf{s}}^{p,q}(D)\). The measure \(\nu_{\Omega}\) is \(\Delta_{\Omega}^{\mathbf{d}}\cdot \mathcal{H}^{m}\) for an appropriate \(\mathbf{d}\) and \(\mathcal{H}^{m}\) the Hausdorff measure. The symbol \(\mathcal{N}\) stands for \(E\times F\) endowed with the group structure \[ (\zeta,x)(\zeta^{'},x^{'}):=(\zeta+\zeta^{'},x+x^{'}+2\operatorname{Im} \Phi(\zeta,\zeta^{'}) \] and \[ f_{h}\colon \mathcal{N}\ni (\zeta,x)\mapsto f(\zeta,x+i\Phi(\zeta)+ih)\in \mathbb{C}. \] The values \(p,q,\mathbf{s}\) for which \(A_{\mathbf{s}}^{p,q}(D)\) is non-trivial are characterized (Proposition 3.5). Some sampling results are obtained (Theorems 3.22 and 3.23). In Section 3.4 the authors deal with atomic decomposition for the spaces \(A_{\mathbf{s}}^{p,q}(D)\). In Section 3.5 there is studied duality of these spaces.
In Chapter 4 the authors introduce and study Besov-type spaces \(B_{p,q}^{\mathbf{s}}(\mathcal{N},\Omega)\). The theory parallels the classical one defined on \(\mathbb{R}^{n}\). It is modelled in relation to the boundary values of the spaces \(A_{\mathbf{s}}^{p,q}(D)\). It should be noted that the group \(\mathcal{N}\) is not commutative and the authors deal with the full range of exponents \(p,q\in (0,\infty]\).
The main results of Chapter 5 concern the boundedness of the Bergman projectors. Theorem 5.25 gives some conditions on \(p,q\in [1,\infty]\) and \(\mathbf{s}, \mathbf{s}^{'}\in \mathbb{R}^{r}\) such that the corresponding projector \(P_{\mathbf{s}^{'}}\) (defined in Definition 5.19) is a continuous linear mapping of \(L_{\mathbf{s},0}^{p,q}(D)\) into \(\tilde{A}_{\mathbf{s}}^{p,q}(D)\) (this space is defined as the image of some extension operator on appropriate Besov spaces – see Definition 5.3). The study of the Bergman projectors is related to atomic decompositions. The chapter opens with a discussion of boundary values \(f_{h}\) as \(h\rightarrow 0\), \(h\in \Omega\) for functions \(f\in A_{\mathbf{s}}^{p,q}(D)\).