Homogeneous Siegel Domains Research Articles

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.

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.

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.

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.

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.

Sharp Norm Estimates for Weighted Bergman Projections in the Mixed Norm Spaces

In this paper, we show that the norm of the Bergman projection on Lp,q-spaces in the upper half-plane is comparable to csc(π/q). Then we extend this result to a more general class of domains, known as the homogeneous Siegel domains of type II.

Lebesgue mixed norm estimates for Bergman projectors: from tube domains over homogeneous cones to homogeneous Siegel domains of type II

In this paper, the weight functions are fundamental compound functions of the homogeneous cone. First, we use the recent Bourgain-Demeter $$l^2$$ -decoupling theorem to obtain the optimal weighted Lebesgue mixed norm estimates for weighted Bergman projectors on tube domains over Lorentz cones. This settles a conjecture of D. Debertol. Secondly, we present a transference principle of weighted Lebesgue mixed norm estimates for weighted Bergman projectors from tube domains over homogeneous cones to homogeneous Siegel domains of type II associated to the same cones. So results of C. Nana for homogeneous Siegel domains of type II can be deduced from earlier results of C. Nana and B. Trojan for tube domains over homogeneous cones. Combining our two theorems, we improve these estimates for homogeneous Siegel domains of type II associated with Lorentz cones, e.g. the Pyateckii-Shapiro Siegel domain of type II.

Biholomorphisms between Hartogs domains over homogeneous Siegel domains

In this paper, we characterize the Hartogs domains over homogeneous Siegel domains of type II and explicitly describe their automorphism groups. Moreover, we prove that any proper holomorphic map between equidimensional Hartogs domains over homogeneous Siegel domains of type II is a biholomorphism.

L p,q -Boundedness of Bergman Projections in Homogeneous Siegel Domains of Type II

In this paper, we generalize to homogeneous Siegel domains of second kind the Lp-continuity properties of the Bergman projection. Precisely, we give an improvement of the index p using Fourier analysis as in the case of convex homogeneous tube type domains (Nana and Trojan in Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) X:477–511, 2011).

Composition operators on the Bergman spaces of a minimal bounded homogeneous domain

Using an integral formula on a homogeneous Siegel domain, we give a necessary and sufficient condition for composition operators on the weighted Bergman space of a minimal bounded homogeneous domain to be compact in terms of a boundary behavior of the Bergman kernel.

A characterization of symmetric Siegel domains by convexity of Cayley transform images

We show that a homogeneous Siegel domain is symmetric if and only if its Cayley transform image is convex. Moreover, this convexity forces the parameter of the Cayley transform to be a specific one, so that the Cayley transform coincides with the inverse of the Cayley transform introduced by Kor´anyi and Wolf.

Admissible wavelets and inverse radon transofrm associated with the affine homogeneous Siegel domains of type II

Let D(Ω,Φ) be the affine homogeneous Siegel domain of type II, whose Silov boundary N is a nilpotent Lie group of step two. In this article, we develop the theory of wavelet analysis on N . By selecting a set of mutual orthogonal wavelets we give a direct sum decomposition of L2(D(Ω,Φ)), the first component A0,0 of which is the Bergman space. Moreover, we study the Radon transform on N , and obtain an inversion formula R−1 = (π)−2dLRL which is an extension of that by Strichartz [R. S. Strichartz, L harmonic analysis and Radon transforms on the Heisenberg group, J. Funct. Anal. 96 (1991), 350–406.]. We devise a subspace of L2(N) on which the Radon transform is a bijection. Using wavelet inverse transform, we establish an inversion formula of the Radon transform in the weak sense.

Asymptotic expansions and Hua-harmonic functions on bounded homogeneous domains

Let D be a homogeneous Siegel domain of type II. We prove that every bounded Hua-harmonic function F on D is pluriharmonic. The proof is based on asymptotic expansion of F.

On symplectic representations of normal j-algebras and their application to Xu's realizations of Siegel domains

We present a canonical construction of an injective normal symplectic representation of a given normal j-algebra. On the other hand, the image of the holomorphic map corresponding to any normal symplectic representation is described as a subset of the Siegel upper half plane by using vector spaces of matrices satisfying certain axioms. Combining these results, we naturally obtain Xu's realization of homogeneous Siegel domains [Y.C. Xu, Automorphism groups of homogeneous bounded domains, Acta Math. Sinica 19 (1976) 161–191; Y.C. Xu, On the isomorphism of homogeneous bounded domains, Acta Math. Sinica 20 (1977) 248–266].

Berezin Transform for Solvable Groups

We study the Berezin transform in the context of solvable groups AN (acting on homogeneous cones and Siegel domains) and determine its spectral decomposition, using an explicit integral kernel representation for the associated ‘eigen-operators’ in terms of multivariable hypergeometric functions.

Geometric norm equality related to the harmonicity of the Poisson kernel for homogeneous Siegel domains

In this paper, we present a geometric norm equality involving an admissible linear form ω for the Shilov boundary of a homogeneous Siegel domain D. We prove that the validity of this norm equality is equivalent to the symmetry of D and the reduction of ω essentially to the Koszul form. This, in particular, reveals a geometric reason that the Poisson kernel is annihilated by the Laplace–Beltrami operator if and only if D is symmetric, a theorem due to Hua, Look, Korányi and Xu.

Family of Cayley transforms of a homogeneous Siegel domain parametrized by admissible linear forms

In this paper we introduce a family of Cayley transforms of a homogeneous Siegel domain D. Each of these transforms is birational and maps D biholomorphically onto a bounded domain in a complex Euclidean vector space.