The Bergman Kernel on Some Hartogs Domains

Abstract

We obtain new explicit formulas for the Bergman kernel function on two families of Hartogs domains. To do so, we first compute the Bergman kernels on the slices of these Hartogs domains with some coordinates fixed, evaluate these kernel functions at certain points off the diagonal, and then apply a first order differential operator to them. We find, for example, explicit formulas for the kernel function onand on {(z 1 , z 2 , w) ∈ C 3 : |z 1 | 2 + |z 2 | 2 + |w| 2 < 1 + |z 2 w| 2 and |w| < 1}.We use our formulas to determine the boundary behavior of the kernel function of these domains on the diagonal. AMS Classification Number: 32A05, 32A07, 32A25, 32A36, 32A40.

$L^p$ estimates for the Bergman projection on some Reinhardt domains

“…In this section we recall a construction from [Huo17], which produces the Bergman kernel of various higher dimensional successors of an initial domain. We start with an initial domain Ω and construct a class of domains U α (Ω) by introducing new parameters α to Ω.…”

“…We start with an initial domain Ω and construct a class of domains U α (Ω) by introducing new parameters α to Ω. The technique in [Huo17] consists of the following 4 steps:…”

“…The main result in [Huo17] relates the Bergman kernel on U α w (Ω) to K U α . To state this result, we need a few more notational definitions.…”

“…In [Huo17], the author began with an initial domain with certain symmetry properties. From this initial domain the author constructed various successor domains and computed (explicitly) the Bergman kernel on them in terms of the Bergman kernel on the initial domain.…”

The holomorphic automorphism groups of twisted Fock-Bargmann-Hartogs domains

“…Recently, this domain was considered by Huo and its Bergman kernel was computed explicitly (see [5]). In contrast to the Fock-Bargmann-Hartogs domains, our argument here does not require an explicit form of the Bergman kernel.…”