Asymptotic Behaviour of Reproducing Kernels of Weighted Bergman Spaces

Let Ω be a domain in Cn, F a nonnegative and G a positive function on Ω such that 1/G is locally bounded, Aα the space of all holomorphic functions on Ω square-integrable with respect to the measure FαGdλ, where dλ is the 2n-dimensional Lebesgue measure, and Kα(x, y) the reproducing kernel for Aα. 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→+∞α x) 1/α = 1/F (x) (∗) holds true. [This fact even plays a crucial role in Berezin’s theory of quantization on curved phase spaces.] In this paper we discuss the validity of this formula in the general case. The answer turns out to depend on, loosely speaking, how well the function − logF can be approximated by certain pluriharmonic functions lying below it. For instance, (∗) holds if − logF is convex (and, hence, can be approximated from below by linear functions), for any function G. Counterexamples are also given to show that in general (∗) may fail drastically, or even be true for some x and fail for the remaining ones. Finally, we also consider the question of convergence of Kα(x, y)1/α for x 6= y, which leads to an unexpected result showing that the zeroes of the reproducing kernels are affected by the smoothness of F : for instance, if F is not real-analytic at some point, then Kα(x, y) must have zeroes for all α sufficiently large.