Asymptotic Behaviour of Reproducing Kernels, Berezin Quantization and Mean-Value Theorems

Abstract

Let Ω be a domain in Cⁿ, F and G positive measurable functions on Ω such that 1/F and 1/G are locally bounded, A ² _α the space of all holomorphic functions on Ω square-integrable with respect to the measure F ^α G dm, where dm is the 2n-dimensional Lebesgue measure, K _α (x,y) the reproducing kernel for A ² _α (if it exists), and ${B_\alpha }f(y) = {K_\alpha }{(y,y)^{ - 1}}\int {_\Omega } f(x){\text{|}}{{\text{K}}_\alpha }{\text{(x,y)}}{{\text{|}}^{\text{2}}}F{(x)^\alpha }G(x)dm(x)$ the Berezin operator on Ω. In this paper we present some results on the asymptotic behavior of K _α and B _α ,as α → +∞. For instance, if − log F is convex then ${\lim _{\alpha \to + \infty }}{K_\alpha }{(x,x)^{1/\alpha }} = 1/F(x)$ for any integrable G, and Kα,(x,y) has a zero for all sufficiently large α whenever F is not real-analytic. Applications to mean value theorems and to quantization on curved phase spaces are also discussed.