Siegel Domain Research Articles

Toeplitz operators on two poly-Bergman-type spaces of the Siegel domain $ D_2 \subset \mathbb{C}^2 $ with continuous nilpotent symbols

<abstract><p>We studied Toeplitz operators acting on certain poly-Bergman-type spaces of the Siegel domain $ D_{2} \subset \mathbb{C}^{2} $. Using continuous nilpotent symbols, we described the $ C^* $-algebras generated by such Toeplitz operators. Bounded measurable functions of the form $ \tilde{c}(\zeta) = c(\text{Im}\, \zeta_{1}, \text{Im}\, \zeta_{2} - |\zeta_1|^{2}) $ are called nilpotent symbols. In this work, we considered symbols of the form $ \tilde{a}(\zeta) = a(\text{Im}\, \zeta_1) $ and $ \tilde{b}(\zeta) = b(\text{Im}\, \zeta_2 -|\zeta_1|^{2}) $, where both limits $ \lim\limits_{s\rightarrow 0^+} b(s) $ and $ \lim\limits_{s\rightarrow +\infty} b(s) $ exist, and $ a $ belongs to the set of piece-wise continuous functions on $ \overline{\mathbb{R}} = [-\infty, +\infty] $ and with one-sided limits at $ 0 $. We described certain $ C^* $-algebras generated by such Toeplitz operators that turned out to be isomorphic to subalgebras of $ M_n(\mathbb{C}) \otimes C(\overline{\Pi}) $, where $ \overline{\Pi} = \overline{ \mathbb{R}} \times \overline{ \mathbb{R}}_+ $ and $ \overline{\mathbb{R}}_+ = [0, +\infty] $.</p></abstract>

Classification of Proper Holomorphic Mappings between Hartogs Domains over Homogeneous Siegel Domains

Classification of Proper Holomorphic Mappings between Hartogs Domains over Homogeneous Siegel Domains

On the theory of Bergman spaces on homogeneous Siegel domains

We consider mixed-norm Bergman spaces on homogeneous Siegel domains. In the literature, two different approaches have been considered and several results seem difficult to be compared. In this paper, we compare the results available in the literature and complete the existing ones in one of the two settings. The results we present are as follows: natural inclusions, density, completeness, reproducing properties, sampling, atomic decomposition, duality, continuity of Bergman projectors, boundary values, and transference.

Lacunary series, resonances, and automorphisms of C2 with a round Siegel domain

Lacunary series, resonances, and automorphisms of C2 with a round Siegel domain

Besov spaces of analytic type: Interpolation, convolution, Fourier multipliers, inclusions

We consider a family of Besov spaces of analytic type on the Šilov boundary N of a homogeneous Siegel domain D, and study their properties concerning convolution, Fourier multipliers, and complex interpolation. In addition, we study how these Besov spaces of analytic type can be compared with the ‘classical’ Besov spaces on N.

Boundedness of Bergman projectors on homogeneous Siegel domains

In this paper we study the boundedness of Bergman projectors on weighted Bergman spaces on homogeneous Siegel domains of Type II. As it appeared to be a natural approach in the special case of tube domains over irreducible symmetric cones, we study such boundedness on the scale of mixed-norm weighted Lebesgue spaces. The sharp range for the boundedness of such operators is essentially known only in the case of tube domains over Lorentz cones. In this paper we prove that the boundedness of such Bergman projectors is equivalent to variuos notions of atomic decomposition, duality, and characterization of boundary values of the mixed-norm weighted Bergman spaces, extending results mostly known only in the case of tube domains over irreducible symmetric cones. Some of our results are new even in the latter simpler context. We also study the simpler, but still quite interesting, case of the “positive” Bergman projectors, the integral operator in which the Bergman kernel is replaced by its modulus. We provide a useful characterization which was previously known for tube domains.

Sharp Hardy-Sobolev-Maz'ya, Adams and Hardy-Adams inequalities on the Siegel domains and complex hyperbolic spaces

The aim of this paper is to establish higher order Poincaré-Sobolev, Hardy-Sobolev-Maz'ya, Adams and Hardy-Adams inequalities on Siegel domains and complex hyperbolic spaces using the method of Helgason-Fourier analysis on the complex hyperbolic spaces. Firstly, we give a factorization theorem for the operators on the complex hyperbolic space which is closely related to Geller's operator, as well as the CR invariant differential operators on the Heisenberg group and CR sphere. (See Theorem 1.3.) Secondly, by using, among other things, the Kunze-Stein phenomenon on a closed linear group SU(1,n) and Helgason-Fourier analysis techniques on the complex hyperbolic spaces, we establish the Poincaré-Sobolev, Hardy-Sobolev-Maz'ya inequality on the Siegel domain Un and the unit ball BCn. (See Theorems 1.4, 1.5.) Finally, we establish the sharp Hardy-Adams inequalities and sharp Adams type inequalities on Sobolev spaces of any positive fractional order on the complex hyperbolic spaces. (See Theorems 1.9, 1.10 and 1.12.) The factorization theorem we prove on the complex hyperbolic spaces is considerably more difficult than in the case of real hyperbolic spaces and should be of its independent interest in the analysis on Heisenberg group and CR sphere and CR invariant differential operators therein.

Some Problems of Complex Analysis in Matrix Siegel Domains

We give a review of recent results in multivariate complex analysis related to matrix Siegel domains.

Morera’s Boundary Theorem in Siegel Domain of the First Kind

The paper considers the realization of the matrix unit polydisk in the form of a Siegel domain of the first kind and proves the boundary analogue of Morera’s theorem

Carleson and Reverse Carleson Measures on Homogeneous Siegel Domains

In this paper we study Carleson and reverse Carleson measures for holomorphic function spaces on a homogeneous Siegel domain of Type II. We prove several necessary conditions and sufficient conditions in order for a measure \(\mu \) to be Carleson and reverse Carleson on mixed-normed weighted Bergman spaces.

Toeplitz and Cesàro-type operators on homogeneous Siegel domains

In this paper we study Toeplitz and Cesàro-type operators on holomorphic function spaces on a homogeneous Siegel domain of Type II. We prove several necessary conditions and sufficient conditions for these operators to be continuous or compact, or to belong to suitable Schatten classes.

Homogeneously Polyanalytic Kernels on the Unit Ball and the Siegel Domain

We prove that the homogeneously polyanalytic functions of total order $m$, defined by the system of equations $\overline{D}^{(k_1,\ldots,k_n)} f=0$ with $k_1+\cdots+k_n=m$, can be written as polynomials of total degree $<m$ in variables $\overline{z_1},\ldots,\overline{z_n}$, with some analytic coefficients. We establish a weighted mean value property for such functions, using a reproducing property of Jacobi polynomials. After that, we give a general recipe to transform a reproducing kernel by a weighted change of variables. Applying these tools, we compute the reproducing kernel of the Bergman space of homogeneously polyanalytic functions on the unit ball in $\mathbb{C}^n$ and on the Siegel domain. For the one-dimensional case, analogous results were obtained by Koshelev (1977), Pessoa (2014), Hachadi and Youssfi (2019).

Gauss-Manin connection in disguise: Genus two curves

We describe an algebra of meromorphic functions on the Siegel domain of genus two which contains Siegel modular forms for an arithmetic index six subgroup of the symplectic group, which is closed under three canonical derivations of the Siegel domain. The main ingredients of our study are the moduli of enhanced genus two curves, Gauss-Manin connection and the modular vector fields living on such moduli spaces.

Holomorphic function spaces on homogeneous Siegel domains

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.

CONNECTION AND CURVATURE ON BUNDLES OF BERGMAN AND HARDY SPACES

We consider a complex domain D x V in the space C-m x C-n and a family of weighted Bergman spaces on V defined by a weight e(-k phi(z , w)) for a pluri-subharmonic function phi(z, w) with a quantization parameter k. The weighted Bergman spaces define an infinite dimensional Hermitian vector bundle over the domain D. We consider the natural covariant differentiation del(z) on the sections, namely the unitary Chern connections preserving the Bergman norm. We prove a Dixmier trace formula for the curvature of the unitary connection and we find the asymptotic expansion for the curvatures R-(k)(Z,Z) for large k and for the induced connection [del((k))(Z), T-f((k))] on Toeplitz operators T-f. In the special case when the domain D is the Siegel domain and the weighted Bergman spaces are the Fock spaces we find the exact formula for [del((k))(Z), T-f((k))] as Toeplitz operators. This generalizes earlier work of J.E. Andersen in Comm. Math. Phys. 255 (2005), 727-745. Finally, we also determine the formulas for the curvature and for the induced connection in the general case of D x V replaced by a general strictly pseudoconvex domain V subset of C-m x C-n fibered over a domain D subset of C-m. The case when the Bergman space is replaced by the Hardy space on the boundary of the domain is likewise discussed.

Algebras Generated by Toeplitz Operators on the Hardy Space over the Siegel Domain

Following the approach of Loaiza and Vasilevski (Equ Oper Theory 92(3): 33, 2020, https://doi.org/10.1007/s00020-020-02580-x ; “Operator Theory, Functional Analysis and Applications”, Oper Theory Adv Appl 28(2), 2020, to appear), we give two different representations of the Hardy space on the Siegel domain in terms of the Bergman space and terms of the direct integral of the weighted Fock spaces. Based on these representations we describe then various commutative and non-commutative $$C^*$$ -algebras, which are generated by Toeplitz operators acting on the Hardy space over the Siegel domain.

Banach algebras generated by Toeplitz operators with parabolic quasi-radial quasi-homogeneous symbols

Let $$D_3$$ be the three-dimensional Siegel domain and $${\mathcal {A}}_\lambda ^2(D_3)$$ the weight-ed Bergman space with weight parameter $$\lambda >-1$$ . In the present paper, we analyse the commutative (not $$C^*$$ ) Banach algebra $${\mathcal {T}}(\lambda )$$ generated by Toeplitz operators with parabolic quasi-radial quasi-homogeneous symbols acting on $${\mathcal {A}}_\lambda ^2(D_3)$$ . We remark that $${\mathcal {T}}(\lambda )$$ is not semi-simple, describe its maximal ideal space and the Gelfand map, and show that this algebra is inverse-closed.

Lie Group Statistics and Lie Group Machine Learning Based on Souriau Lie Groups Thermodynamics & Koszul-Souriau-Fisher Metric: New Entropy Definition as Generalized Casimir Invariant Function in Coadjoint Representation.

In 1969, Jean-Marie Souriau introduced a “Lie Groups Thermodynamics” in Statistical Mechanics in the framework of Geometric Mechanics. This Souriau’s model considers the statistical mechanics of dynamic systems in their “space of evolution” associated to a homogeneous symplectic manifold by a Lagrange 2-form, and defines in case of non null cohomology (non equivariance of the coadjoint action on the moment map with appearance of an additional cocyle) a Gibbs density (of maximum entropy) that is covariant under the action of dynamic groups of physics (e.g., Galileo’s group in classical physics). Souriau Lie Group Thermodynamics was also addressed 30 years after Souriau by R.F. Streater in the framework of Quantum Physics by Information Geometry for some Lie algebras, but only in the case of null cohomology. Souriau method could then be applied on Lie groups to define a covariant maximum entropy density by Kirillov representation theory. We will illustrate this method for homogeneous Siegel domains and more especially for Poincaré unit disk by considering SU(1,1) group coadjoint orbit and by using its Souriau’s moment map. For this case, the coadjoint action on moment map is equivariant. For non-null cohomology, we give the case of Lie group SE(2). Finally, we will propose a new geometric definition of Entropy that could be built as a generalized Casimir invariant function in coadjoint representation, and Massieu characteristic function, dual of Entropy by Legendre transform, as a generalized Casimir invariant function in adjoint representation, where Souriau cocycle is a measure of the lack of equivariance of the moment mapping.

Weighted Integral Representations of Pluriharmonic Functions in the Siegel Domain of $$C^n$$

The Siegel domain in the space $$C^n$$ is defined as follows: $$\begin{aligned} \Omega _n=\left\{ \eta =(\eta _1,\eta _2,\dots ,\eta _n)\in C^n: Im \eta _1>\sum _{k=2}^n |\eta _k|^2 \right\} . \end{aligned}$$In the paper the weighted spaces $$L_{\alpha }^p(\Omega _n)$$ with the weight function of the type $$(Im\eta _1-\sum _{k=2}^n |\eta _k|^2)^{\alpha }$$ are introduced. For pluriharmonic functions u from the spaces $$L_{\alpha }^p(\Omega _n)$$ weighted integral representations are established. Inequalities between weighted $$L^p$$-norms of conjugate pluriharmonic functions are shown.

Radon transforms on Siegel-type nilpotent Lie groups

Let $$\mathfrak{N}:=H_n\times\mathbb{C}^n$$ be the Siegel-type nilpotent group, which can be identified as the Shilov boundary of Siegel domain of type II, where Hn denotes the set of all n × n Hermitian matrices. In this article, we use singular convolution operators to define Radon transform on $$\mathfrak{N}$$ and obtain the inversion formulas of Radon transforms. Moveover, we show that Radon transform on $$\mathfrak{N}$$ is a unitary operator from Sobolev space Wn,2 into $$L^2(\mathfrak{N})$$ .