Asymptotic behaviour of reproducing kernels of weighted Bergman spaces

Abstract

Abstract. Let Ω be a domain in C n , F a nonnegative and G a positive function on Ω such that 1/G is locally bounded, A 2 α the space of all holomorphic functions on Ω square-integrable with respect to the measure F α G dλ, where dλ is the 2n-dimensional Lebesgue measure, and Kα(x, y) the reproducing kernel for A 2 α . It has been known for a long time that in some special situations (such as on bounded symmetric domains Ω with G = 1 and F = the Bergman kernel function) the formula lim α→+∞

High-power asymptotics of some weighted harmonic Bergman kernels

“…, thus recovering, in particular, the cases x = 0 in ( 6) and (8). Note that not only the leading power of α is now different than for x = 0 (α n/2 opposed to α n−1 ), but also the powers go down not by 1 but by 1 2 in the full expansion.…”

Vanishing Bergman kernels on the disk

“…The vanishing of Bergman kernels on domains of C n is known as the Lu Qi-Keng's problem [12]. It has been studied extensively by H. P. Boas, S. Fu, E. J. Straube, M. Englis and others; we list here the following papers: [2], [3], [4] and [7]. We also mention the paper of Y. E. Zeytuncu [18], where a weight λ on D is constructed so that λ ∼ 1 and B λ z has a zero.…”