Szego Kernel Research Articles

О сходимости порядкосохраняющего слабого жадного алгоритма для подпространств, порожденных ядром Сегё в пространстве Харди

О сходимости порядкосохраняющего слабого жадного алгоритма для подпространств, порожденных ядром Сегё в пространстве Харди

An explicit formula of Cauchy-Szego kernel for quaternionic Siegel upper half space and applications

In this paper we obtain an explicit formula of Cauchy--Szego kernel for quaternionic Siegel upper half space, and then based on this, we prove that the Cauchy--Szego projection on quaternionic Heisenberg group is a Calderon--Zygmund operator via verifying the size and regularity conditions for the kernel. Next, we also obtain a suitable version of pointwise lower bound for the kernel, which further implies the characterisations of the boundedness and compactness of commutator of the Cauchy--Szego operator via the BMO and VMO spaces on quaternionic Heisenberg group, respectively.

On Weights Which Admit Reproducing Kernel of Szegő Type

In this paper we generalize the concept of the Szego kernel by putting the weight of integration in the definition of the inner product in the Szego space. We give some sufficient conditions for the weight in order for the Szego kernel of the correspoding space to exist. We give examples of weights on unit ball for which there is no Szego kernel of the corresponding Szego space. Then using biholomorphisms we prove that there exist such weights for a large class of domains. At the end we show that weighted Szego kernel depends continuously in some sense on weight of integration.

ON A CONSTRUCTION OF A FRAME IN THE HARDY SPACE DEFINED ON THE TWO-DIMENSIONAL POLYDISC

In the article we present a construction of a representing system based on the discretized Szego kernel in the Hardy space defined on the two-dimensional polydisc. An answer to the question on the existence of representing systems based on reproducing kernels depends significantly on the space under consideration. It is well known that in the Hardy space there are no both bases and Duffin Shaeffer frames, based on the discretized Szego kernel. We use a notion of a Banach frame which generalizes the concept of the Duffin Shaeffer frame. Having constructed a Banach frame, we can say that any function from the Hardy space can be represented as a series of discretized kernels.

Existence of Frames Based on the Szegö Kernel in the Hardy Space

As is known, a sequence of functions consisting of meanings of the Szego reproducing kernel of the Hardy space in the unit disk cannot be Duffin—Schaeffer frame. In the present paper we show that the using of more general conception of frame enables us to solve positively the question on existence of a generalized frame built in terms of the Szego kernel.

Geometric Formula for Prime form on a Sewn Riemann Surface

We use the geometric representation for the Szego kernel on genus g1+g2 Riemann surface obtained in the frames of the Yamada sewing construction of two Riemann surfaces of genus g1 and genus g2 in order to derive new formulas expressing prime forms. These formulas can be used in vertex operator algebra and conformal field theory computation as well as in algebraic geometry.

Local Scaling Asymptotics for the Gutzwiller Trace Formula in Berezin–Toeplitz Quantization

Let f be a smooth function on a quantizable compact Kahler manifold, with the special property that the associated Hamiltonian flow acts by isometries. We consider a naturally associated Berezin–Toeplitz operator, which generates a unitary action on the Hardy space of the quantization. In this setting, we produce local scaling asymptotics in the semiclassical regime for a Berezin–Toeplitz version of the Gutzwiller trace formula, in the spirit of the near-diagonal scaling asymptotics of Szego and Toeplitz kernels. More precisely, we consider an analogue of the ‘Gutzwiller–Toeplitz kernel’ previously introduced in this setting by Borthwick, Paul, and Uribe, and study how it asymptotically concentrates along the appropriate classical loci defined by the dynamics, with an explicit description of the exponential decay along normal directions. These local scaling asymptotics probe into the concentration behavior of the eigenfunctions of the quantized Hamiltonian flow. When globally integrated, they yield the analogue of the Gutzwiller trace formula.

Morse inequalities for Fourier components of Kohn–Rossi cohomology of CR manifolds with $$S^1$$ S 1 -action

Let X be a compact connected CR manifold of dimension \(2n-1, n\ge 2\) with a transversal CR \(S^1\)-action on X. We study the Fourier components of the Kohn–Rossi cohomology with respect to the \(S^1\)-action. By studying the Szego kernel of the Fourier components we establish the Morse inequalities on X. Using the Morse inequalities we have established on X we prove that there are abundant CR functions on X when X is weakly pseudoconvex and strongly pseudoconvex at a point.

A Comparison Between the Bergman and Szegö Kernels of the Non-smooth Worm Domain $$D'_{\upbeta }$$ D β ′

In this work we provide an asymptotic expansion for the Szego kernel associated to a suitably defined Hardy space on the non-smooth worm domain $$D'_{\upbeta }$$ . After describing the singularities of the kernel, we compare it with an asymptotic expansion of the Bergman kernel. In particular, we show that the Bergman kernel has the same singularities of the first derivative of the Szego kernel with respect to any of the variables. On the side, we prove the boundedness of the Bergman projection operator on Sobolev spaces of integer order.

FORELLI–RUDIN CONSTRUCTION AND ASYMPTOTIC EXPANSION OF SZEGÖ KERNEL ON REINHARDT DOMAINS

We apply Forelli--Rudin construction and Nakazawa's hodograph transformation to prove a graph theoretic closed formula for invariant theoretic coefficients in the asymptotic expansion of the Szego kernel on strictly pseudoconvex complete Reinhardt domains. The formula provides a structural analogy between the asymptotic expansion of the Bergman and Szego kernels. It can be used to effectively compute the first terms of Fefferman's asymptotic expansion in CR invariants. Our method also works for the asymptotic expansion of the Sobolev--Bergman kernel introduced by Hirachi and Komatsu.

The Weighted Super Bergman Kernels Over the Supermatrix Spaces

The purpose of this paper is threefold. Firstly, using Howe duality for \((\mathfrak {gl}(m_{1}|n_{1}),\mathfrak {gl}(m_{2}|n_{2}))\), we obtain integral formulas of the super Schur functions with respect to the super standard Gaussian distributions. Secondly, we give explicit expressions of the super Szego kernels and the weighted super Bergman kernels for the Cartan superdomains of type I. Thirdly, combining these results, we obtain duality relations of integrals over the unitary groups and the Cartan superdomains, and the marginal distributions of the weighted measure.

Approximation of monogenic functions by higher order Szegö kernels on the unit ball and half space

We study the adaptive decomposition of functions in the monogenic Hardy spaces \(\mathcal{H}^2\) by higher order Szego kernels under the framework of Clifford algebra and Clifford analysis, in the context of unit ball and half space. This is a sequel and a higher-dimensional generalization of our recent study on the complex Hardy spaces.

On Octonionic Regular Functions and the Szegö Projection on the Octonionic Heisenberg Group

We discuss the octonionic regular functions and the octonionic regular operator on the octonionic Heisenberg group. This is the octonionic version of CR function theory in the theory of several complex variables and regular function theory on the quaternionic Heisenberg group. By identifying the octonionic algebra with \(\mathbb{R }^{8}\), we can write the octonionic regular operator and the associated Laplacian operator as real \((8\times 8)\)-matrix differential operators. Then we use the group Fourier transform on the octonionic Heisenberg group to analyze the associated Laplacian operator and to construct its kernel. This kernel is exactly the Szego kernel of the orthonormal projection from the space of \(L^{2}\) functions to the space of \(L^{2}\) regular functions on the octonionic Heisenberg group.

Szegö kernel, regular quantizations and spherical CR-structures

We compute the Szego kernel of the unit circle bundle of a negative line bundle dual to a regular quantum line bundle over a compact Kahler manifold. As a corollary we provide an infinite family of smoothly bounded strictly pseudoconvex domains on complex manifolds (disk bundles over homogeneous Hodge manifolds) for which the log-terms in the Fefferman expansion of the Szego kernel vanish and which are not locally CR-equivalent to the sphere. We also give a proof of the fact that, for homogeneous Hodge manifolds, the existence of a locally spherical CR-structure on the unit circle bundle alone implies that the manifold is biholomorphic to a projective space. Our results generalize those obtained by Englis (Math Z 264(4):901–912, 2010) for Hermitian symmetric spaces of compact type.

On the Cauchy–Szegö Kernel for Quaternion Siegel Upper Half-Space

The work is dedicated to the construction of the Cauchy–Szego kernel for the Cauchy–Szego projection integral operator from the space of \(L^2\)-integrable functions defined on the boundary of the quaternion Siegel upper half-space to the space of boundary values of the quaternion regular functions of the Hardy space over the quaternion Siegel upper half-space.

Higher Genus Characters for Vertex Operator Superalgebras on Sewn Riemann Surfaces

We review our recent results on computation of the higher genus characters for vertex operator superalgebras modules. The vertex operator formal parameters are associated to local parameters on Riemann surfaces formed in one of two schemes of (self- or tori- ) sewing of lower genus Riemann surfaces. For the free fermion vertex operator superalgebra we present a closed formula for the genus two continuous orbifold partition functions (in either sewings) in terms of an infinite dimensional determinant with entries arising from the original torus Szego kernel. This partition function is holomorphic in the sewing parameters on a given suitable domain and possesses natural modular properties. Several higher genus generalizations of classical (including Fay’s and Jacobi triple product) identities show up in a natural way in the vertex operator algebra approach.

Determinantal characterization of canonical curves and combinatorial theta identities

We characterize genus g canonical curves by the vanishing of combinatorial products of g+1 determinants of Brill-Noether matrices. This also implies the characterization of canonical curves in terms of (g-2)(g-3)/2 theta identities. A remarkable mechanism, based on a basis of H^0(K_C) expressed in terms of Szego kernels, reduces such identities to a simple rank condition for matrices whose entries are logarithmic derivatives of theta functions. Such a basis, together with the Fay trisecant identity, also leads to the solution of the question of expressing the determinant of Brill-Noether matrices in terms of theta functions, without using the problematic Klein-Fay section sigma.

The reproducing kernels and the finite type conditions

We discuss and expand upon our previous results on comparison of the Bergman and the Szego kernels. We also study the relations between the Bergman growth exponent and the finite type conditions.

The Szegö Kernel and Proper Holomorphic Mappings to a Half Plane

We prove that all proper holomorphic mappings from a finitely connected domain in the plane to the right half plane can be expressed simply in terms of the Szego kernel associated to the domain. Our decomposition also reveals the linear structure of the semi-group of all such maps and it offers a method to construct proper holomorphic maps of arbitrary mapping degree.

On the construction of uniformly convergent disk polynomial expansions

We present a method to derive explicit uniformly convergent disk polynomial expansions for real-analytic functions on the complex unit disk. As an application, we deduce the spherical harmonic expansion of the Poisson–Szego kernel for the unit ball in \({\mathbb{C}^q}\) .