New classes of domains with explicit Bergman kernel

We introduce two classes of egg type domains, built on general bounded sym-metric domains, for which we obtain the Bergman kernel in explicit formulas. As an aux-iliary tool, we compute the integral of complex powers of the generic norm on a boundedsymmetric domains using the well-known integral of Selberg. This generalizes matrix in-tegrals of Hua and leads to a special polynomial with integer or half-integer coefficientsattached to each irreducible bounded symmetric domain.     

An explicit computation of the Bergman kernel function

We give an explicit computation of the Bergman kernel function on the domain

The Bergman kernel function of some Reinhardt domains

The boundary behavior of the Bergman Kernel function of some Reinhardt domains is studied. Upper and lower bounds for the Bergman kernel function are found at the diagonal points . Let be the Reinhardt domain

where , ; and let be the Bergman kernel function of . Then there exist two positive constants and and a function such that

holds for every . Here

and is the defining function for . The constants and depend only on and , not on .

     

The Bergman kernel function

In this note, we point out that a large family of n×n matrix valued kernel functions defined on the unit disc $ \mathbb{D} \subseteq \mathbb{C} $ \mathbb{D} \subseteq \mathbb{C} , which were constructed recently in [9], behave like the familiar Bergman kernel function on $ \mathbb{D} $ \mathbb{D} in several different ways. We show that a number of questions involving the multiplication operator on the corresponding Hilbert space of holomorphic functions on $ \mathbb{D} $ \mathbb{D} can be answered using this likeness.     

Zeros of Bergman kernels on some Hartogs domains

Explicit Bergman kernels are obtained on some Hartogs domains. For some special cases, zeros of the kernels are considered.     

A note on the Bergman kernel of a certain Hartogs domain

We give an explicit formula of the Bergman kernel of a certain Hartogs domain.     

The Bergman kernel functions on Hua domains

We get the Bergman kernel functions in explicit formulas on four types of Hua domain.There are two key steps: First, we give the holomorphic automorphism groups of four types of Hua domain; second, we introduce the concept of semi-Reinhardt domain and give their complete orthonormal systems. Based on these two aspects we obtain the Bergman kernel function in explicit formulas on Hua domains.     

The Bergman kernel and the green function

Definition of the Bergman space for an arbitrary operator is given. Sufficient conditions for the existence of the Bergman kernel for this space are obtained. For an elliptic operator, the Bergman kernel is represented via the Green function. Bibliography: 12 titles. Dedicated to N. N. Uraltseva on her jubilee Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 221, 1995, pp. 145–166. Translated by S. Yu. Pilyugin.     

Estimates of the Bergman kernel for some pseudoconvex domains

Denote by K_ω(z, ζ) the Bergman kernel of a pseudoconvex domain Ω. For some classes of domains Ω, a relationship is found between the rate of increase of K_ω(z, z) as z tends to ∂Ω, and a purely geometric property of Ω. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 222, 1995, pp. 222–245.     

The Bergman kernel of the symmetrized polydisc in higher dimensions has zeros

We prove that the Bergman kernel of the symmetrized polydisc in dimension greater than two has zeros. Received: 15 November 2005; revised: 10 January 2006     

Bergman kernel function on Hua Construction of the second type

In this paper, we give an explicit formula of the Bergman kernel function on Hua Construction of the second type when the parameters 1/p1,…,1/(pr-1) are positive integers and 1/pr is an arbitrary positive real number.     

Bergman kernel function on Hua Construction of the second type

In this paper, we give an explicit formula of the Bergman kernel function on Hua Construction of the second type when the parameters 1/p1, ..., 1/p_r−1 are positive integers and 1/p_r is an arbitrary positive real number.

     

Bergman kernel and metric on non-smooth pseudoconvex domains

A stability theorem of the Bergman kernel and completeness of the Bergman metric have been proved on a type of non-smooth pseudoconvex domains defined in the following way:D = {z∈U|r(z)} <whereU is a neighbourhood of andr is a continuous plurisubharmonic function onU. A continuity principle of the Bergman Kernel for pseudoconvex domains with Lipschitz boundary is also given, which answers a problem of Boas.