Berezin kernels and maximal degenerate representations associated with Riemannian symmetric spaces of Hermitian type

We introduce the Berezin kernels for Riemannian symmetric spaces of Hermitian type by restricting the maximal degenerate representations of the corresponding noncompactly causal Lie groups. Bibliography: 20 titles.Published in Zapiski Nauchnykh Seminarov POMI, Vol. 292, 2002, pp. 11–21.This revised version was published online in April 2005 with a corrected cover date and article title.     

On the separation of spectra in the analysis of Berezin kernels

The problem of restricting a highest weight representation of the group U(p, q) to the subgroup O(p, q) is considered. This restriction has an intricate spectrum that contains representations of different types. We construct a decomposition of this representation into reducible representations each of which has a single-type spectrum. Some integrals over classical groups are also calculated; these integrals generalize those of Hua. Partially supported by grant RFBR 98-01-00303 and the Russian program of support of leading scientific schools (grant RFBR 96-01-96249). Moscow Institute of Electronics and Mathematics. Translated from Funktsional'nyi analiz i Ego Prilozheniya, Vol. 34, No. 3, pp. 49–62, July–September, 2000. Translated by Yu. A. Neretin     

Berezin transform and Toeplitz operators on harmonic Bergman spaces

For an operator which is a finite sum of products of finitely many Toeplitz operators on the harmonic Bergman space over the half-space, we study the problem: Does the boundary vanishing property of the Berezin transform imply compactness? This is motivated by the Axler-Zheng theorem for analytic Bergman spaces, but the answer would not be yes in general, because the Berezin transform annihilates the commutator of any pair of Toeplitz operators. Nevertheless we show that the answer is yes for certain subclasses of operators. In order to do so, we first find a sufficient condition on general operators and use it to reduce the problem to whether the Berezin transform is one-to-one on related subclasses.     

Berezin transform in polynomial bergman spaces

Fix a smooth weight function Q in the plane, subject to a growth condition from below. Let K_m,n denote the reproducing kernel for the Hilbert space of analytic polynomials of degree at most n ? 1 of finite L²‐norm with respect to the measure e^?mQ dA. Here dA is normalized area measure, and m is a positive real scaling parameter. The (polynomial) Berezin measure for the point z₀ is a probability measure that defines the (polynomial) Berezin transform for continuous . We analyze the semiclassical limit of the Berezin measure (and transform) as m → + ∞ while n = m τ + o(1), where τ is fixed, positive, and real. We find that the Berezin measure for z₀ converges weak‐star to the unit point mass at the point z₀ provided that Δ Q(z₀) > 0 and that z₀ is contained in the interior of a compact set , defined as the coincidence set for an obstacle problem. As a refinement, we show that the appropriate local blowup of the Berezin measure converges to the standardized Gaussian measure in the plane. For points , the Berezin measure cannot converge to the point mass at z₀. In the model case Q(z) = |z|², when is a closed disk, we find that the Berezin measure instead converges to harmonic measure at z₀ relative to . Our results have applications to the study of the eigenvalues of random normal matrices. The auxiliary results include weighted L²‐estimates for the equation when f is a suitable test function and the solution u is restricted by a polynomial growth bound at ∞. © 2009 Wiley Periodicals, Inc.     

Berezin transform on¶compact Hermitian symmetric spaces

Let X=G ^* be a compact Hermitian symmetric space. We study the Berezin transform on L ²(X) and calculate its spectrum under the decomposition of L ²(X) into the irreducible representations of G ^*. As applications we find the expansion of powers of the canonical polynomial (Bergman reproducing kernel for the canonical line bundle) in terms of the spherical polynomials on X, and we find the irreducible decomposition of tensor products of Bergman spaces on X. Received: 10 September 1996 / Revised version: 10 September 1997     

Berezin transform on harmonic Bergman spaces on the real ball

We provide the full asymptotic expansion of the harmonic Berezin transform on the unit ball in Rⁿ${\mathbb{R}}^n$ purely by means of transformations of hypergeometric functions and function?s “hypergeometrization”.     

Berezin symbol and invertibility of operators on the functional Hilbert spaces

We give in terms of reproducing kernel and Berezin symbol the sufficient conditions ensuring the invertibility of some linear bounded operators on some functional Hilbert spaces.     

Reproducing kernel Hilbert spaces associated with kernels on topological spaces

We analyze reproducing kernel Hilbert spaces of positive definite kernels on a topological space X being either first countable or locally compact. The results include versions of Mercer’s theorem and theorems on the embedding of these spaces into spaces of continuous and square integrable functions.     

变量核分数次积分在Hardy空间的有界性

§ 1　 Introduction and main resultsLet Sn- 1 be the unitsphere in Rn(n≥ 2 ) equipped with normalized Lebesgue measure dσ= dσ(z′) .We say that a functionΩ(x,z) defined on Rn× Rnbelongs to L∞ (Rn)× Lr(Sn- 1 )(r≥ 1 ) ,ifΩ(x,z) satisfies the following two conditions,(i) for any x,z∈Rnandλ>0 ,there hasΩ(x,λz) =Ω(x,z) ;(ii)‖Ω‖L∞(Rn)× Lr(Sn- 1) :=supx∈ Rn∫Sn- 1|Ω(x,z′) | rdσ(z′) 1 / r<∞ .For 0 <α相似文献   

Minimal kernels of weakly complete spaces

Let X be a weakly complete space i.e. X a complex space endowed with a C^k-smooth, k?0, plurisubharmonic exhaustion function. We give the notion of minimal kernelΣ¹=Σ¹(X) of X by the following property: x∈Σ¹ if no continuous plurisubharmonic exhaustion function is strictly plurisubharmonic near x. The study of the geometric properties of the minimal kernels is the aim of present paper. After stating that the minimal kernel Σ¹ of a weakly complete space can be defined by a single plurisubharmonic exhaustion function ?, called minimal, using the characterization in terms of Bremermann envelopes, we prove the following, crucial, result: if X is a weakly complete manifold and ? a minimal function for X, the nonempty level sets Σ_c¹=Σ¹∩{?=c} have the local maximum property. In the last section we discuss the special case of weakly complete surfaces. We prove that if dim_cX=2 and c is a regular value of a minimal function ? then the nonempty level sets Σ_c¹=Σ¹∩{?=c} are compact spaces foliated by holomorphic curves.     

Reproducing kernels and contractive divisors in bergman spaces

In the Hardy spaces H^p of holomorphic functions, Blaschke products are applied to factor out zeros. However, for Bergman spaces, the zero sets of which do not necessarily satisfy the Blaschke condition, the study of divisors is a more recent development. Hedenmalm proved the existence of a canonical contractive zero-divisor which plays the role of a Blascke product in the Bergman space . Duren, Khavinson, Shapiro, and Sundberg later extended Hedenmalm's result to , 0<p<∞. In this paper, an explicit formula for the contractive divisor is given for a zero set that consists of two points of arbitrary multiplicities. There is a simple one-to-one correspondence between contractive divisors and reproducing kernels for certain weighted Bergman spaces. The divisor is obtained by calculating the associated reproducing kernel. The formula is then applied to obtain the contractive divisor for a certain regular zero set, as well as the contractive divisor associated with an inner function that has singular support on the boundary. Bibliography: 13 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 232, 1996, pp. 174–198.     

Analytic continuation of resolvent kernels on noncompact symmetric spaces

For certain real hypersurfaces in the projective space, of signature (1,n), we study the filling problem for small deformations of the CR structure (the other signatures being well understood). We characterize the deformations which are fillable, and prove that they have infinite codimension in the set of all CR structures. This result generalizes the cases of the 3-sphere and of signature (1,1) to higher dimension.The author is a member of EDGE, Research Training Network, HPRN-CT-2000-00101, supported by the European Human Potential Programme.     

Integral operators with variable kernels on weak Hardy spaces

In this note the authors study the mapping properties of a class of integral operators with variable kernels on the weak Hardy spaces.     

Reproducing kernels for Hardy spaces on multiply connected domains

Associated with a boundedg-holed (g0) planar domainD are two types of reproducing kernel Hilbert spaces of meromorphic functions onD. We give explicit formulas for the reproducing kernel functions of these spaces. The formulas are in terms of theta functions defined on the Jacobian variety of the Schottky double of the regionD. As applications we settle a conjecture of Abrahamse concerning Nevalinna-Pick interpolation on an annulus and obtain explicit formulas for the curvature (in the sense of Cowen and Douglas) of rank 1 bundle shift operators.     

Radial kernels and their reproducing kernel Hilbert spaces

We describe how to use Schoenberg’s theorem for a radial kernel combined with existing bounds on the approximation error functions for Gaussian kernels to obtain a bound on the approximation error function for the radial kernel. The result is applied to the exponential kernel and Student’s kernel. To establish these results we develop a general theory regarding mixtures of kernels. We analyze the reproducing kernel Hilbert space (RKHS) of the mixture in terms of the RKHS’s of the mixture components and prove a type of Jensen inequality between the approximation error function for the mixture and the approximation error functions of the mixture components.     

Bases of reproducing kernels in de-Branges spaces

This paper deals with geometric properties of sequences of reproducing kernels related to de-Branges spaces. If b is a nonconstant function in the unit ball of H^∞, and T_b is the Toeplitz operator, with symbol b, then the de-Branges space, H(b), associated to b, is defined by , where H² is the Hardy space of the unit disk. It is equipped with the inner product such that is a partial isometry from H² onto H(b). First, following a work of Ahern-Clark, we study the problem of orthogonal basis of reproducing kernels in H(b). Then we give a criterion for sequences of reproducing kernels which form an unconditional basis in their closed linear span. As far as concerns the problem of complete unconditional basis in H(b), we show that there is a dichotomy between the case where b is an extreme point of the unit ball of H^∞ and the opposite case.     

Heat kernels on metric spaces and a characterisation of constant functions

Using the Dirichlet forms theory, we prove that when the locally compact measure metric space (X,,m) is the state space of a Markov process with transition density p(t,x,y) bounded from the above by where is a function satisfying certain integrability condition, then the following statement holds: when u L²(X) and then u is a constant function.This work is partially supported by a KBN grant no. 2-PO3A-028-22.Mathematics Subject Classification (2000): primary 60J35, secondary 46E35