The Bergman kernel functions of certain unbounded domains

Abstract

Abstract. We compute the Bergman kernel functions of the unbounded domainsIt is also shown that these kernel functions have no zeros in Ωp. We use a method from harmonic analysis to reduce the computation of the 2-dimensional case to the problem of finding the kernel function of a weighted space of entire functions in one complex variable.

Non-analytic Bergman and Szegö kernels for weakly pseudoconvex tube domains in $\mathbb C^2$

“…4). In the case of the Szegö kernel, similar integral representations were obtained in [14], [9], [17], etc., so Theorem 1.2 can be shown in a similar fashion (Sect. 5).…”

“…It is known in [14], [9], [17] that the Szegö kernel of a tube domain R 2 +iω is also represented as follows:…”

Newton polyhedra and the Bergman kernel

“…We remark that the function τ → K(z; τ ) is continuous for fixed z from the result in [14]. Haslinger [21], [22] obtained an interesting relation between K(z; τ ) and the Bergman kernel…”

“…Our analysis is based on some integral formula for the Bergman kernel due to Haslinger [21], [22]. Under weak assumptions on a domain in C n+1 , the Fourier transform gives a clear connection between the Bergman space of and an associated weighted Bergman space H τ in C n with a parameter τ .…”

The □ b -Heat Equation on Quadric Manifolds

“…, z n ) = n j =1 p j (z j ), and p j are subharmonic, nonharmonic polynomials, Raich [14][15][16][17] has estimated the heat kernel associated with the weighted ∂-problem. If, in addition, n = 1, the weighted ∂-problem and explicit construction of Bergman and Szegö kernels have been studied by a number of authors in different contexts, e.g., [1,6,[8][9][10][11]. We also note that quadric manifolds are related to H -type groups on which Yang and Zhu have computed the heat kernel for the sub-Laplacian [20].…”