On the Bergman Kernel of Hyperconvex Domains Takeo Ohsawa

Mathematische Zeitschrift, 1996

Let Ω be a bounded pseudoconvex domain in C n with smooth defining function r and let z 0 ∈ bΩ be a point of finite type. We also assume that the Levi form ∂∂r(z) of bΩ has (n − 2)positive eigenvalues at z 0. Then we estimate the Bergman kernel function and its derivatives off the diagonal near z 0. 2000 Mathematics Subject Classification. 32H15.

Proceedings of the American Mathematical Society

The boundary behavior of the Bergman metric near a convex boundary point z 0 of a pseudoconvex domain D ⊂ C n is studied; it turns out that the Bergman metric at points z ∈ D in direction of a fixed vector X 0 ∈ C n tends to infinite, when z is approaching z 0 , if and only if the boundary of D does not contain any analytic disc through z 0 in direction of X 0 .

Studia Mathematica, 2002

We give a characterization of L 2 h -domains of holomorphy with the help of the boundary behavior of the Bergman kernel and geometric properties of the boundary, respectively.

Bulletin of the American Mathematical Society, 1981

If D is a bounded open subset of C", the set H = {ƒ: D-> C| ƒ is holomorphic and S D \f\ 2 < +°°} is a separable infinite-dimensional Hubert space relative to the inner product <ƒ, g) = f D fg. The completeness of H can be seen from Cauchy integral estimates. Similar estimates show that for any p E D the functional ƒ H* ƒ(/?),ƒ£ H, is continuous. Thus there is a unique element K D (z, p) E f/ (as a function of z) such that, f(p) = f f(z)K D (z, p)dV(z) for all ƒ G H. The function K D is called the Bergman kernel function. If {y i }™ = , l is an orthonormal basis for f/ then K D (z, p) = ^.^.(z)^/?). The convergence of the series is absolute, uniformly on compact subsets ofD x D. For any z ED, K D (z, z) > 0 and log K D (z, z) is a real analytic function on D. The Hermitian form 3 2 ^âTÂF log K D(Z > z^dz i 0 dz~j i,j oz i oz j

Complex Analysis in Several Variables — Memorial Conference of Kiyoshi Oka's Centennial Birthday, Kyoto/Nara 2001

Eprint Arxiv Math 9909164, 1999

In the paper we study the problems of the boundary behaviour of the Bergman kernel and the Bergman completeness in some classes of bounded pseudoconvex domains, which contain also non-hyperconvex domains. Among the classes for which we prove the Bergman completeness and the convergence of the Bergman kernel to infinity while tending to the boundary are all bounded pseudonvex balanced domains, all bounded Hartogs domains with balanced fibers over regular domains and some bounded Laurent-Hartogs domains.

arXiv: Complex Variables, 2020

We construct a pointwise Boutet de Monvel-Sjostrand parametrix for the Szegő kernel of a weakly pseudoconvex three dimensional CR manifold of finite type assuming the range of its tangential CR operator to be closed; thereby extending the earlier analysis of Christ. This particularly extends Fefferman&#39;s boundary asymptotics of the Bergman kernel to weakly pseudoconvex domains in $\mathbb{C}^{2}$, in agreement with D&#39;Angelo&#39;s example. Finally our results generalize a three dimensional CR embedding theorem of Lempert.

Monatsh Math, 2003

The nontangential behavior of the Bergman metric near a smooth convex boundary point of a bounded pseudoconvex domain D & C n is studied in terms of its multitype.

1997

The hyperholomorphic Bergmann kernel function *B for a domain R is introduced as the special quaternionic "derivative" of the Green function for R. It is shown that *B is hyperholomorphic, Hermitian symmetric and reproduces hyperholornorphic functions. We obtain an integral representation of *B as a sum of two integrals. One of them gives a smooth function, and the other describes the behaviour of +B near a boundary. To investigate the hyperholomorphic Bergmann function for some fixed class of hyperholornorphic functions we have to use not only the properties of just this class but also those of some other classes. The second fact is completely unpredictable from the complex analysis point of view. The connection between the hyperholomorphic Bergmann projector

The purpose of this paper is to prove $L^p$-Sobolev and H\&quot;older estimates for the Bergman projection on both pseudoconvex domains of finite type and a large class of pseudoconvex domains of infinite type.