Asymptotic behaviour of reproducing kernels
of weighted Bergman spaces
Author: Miroslav Englis
Journal: Trans. Amer. Math. Soc. 349 (1997), 3717-3735
MSC (1991): Primary 30C40, 32H10; Secondary 31C10, 30E15
MathSciNet review: 1401769
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Abstract: Let be a domain in , a nonnegative and a positive function on such that is locally bounded, the space of all holomorphic functions on square-integrable with respect to the measure , where is the -dimensional Lebesgue measure, and the reproducing kernel for . It has been known for a long time that in some special situations (such as on bounded symmetric domains with and the Bergman kernel function) the formula
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 can be approximated by certain pluriharmonic functions lying below it. For instance, () holds if is convex (and, hence, can be approximated from below by linear functions), for any function . Counterexamples are also given to show that in general () may fail drastically, or even be true for some and fail for the remaining ones. Finally, we also consider the question of convergence of for , which leads to an unexpected result showing that the zeroes of the reproducing kernels are affected by the smoothness of : for instance, if is not real-analytic at some point, then must have zeroes for all sufficiently large.
Affiliation: Mathematical Institute of the Academy of Sciences, Žitná 25, 11567 Prague 1, Czech Republic
Keywords: Bergman space, reproducing kernels, asymptotic behaviour, lower pluriharmonic envelopes, plurisubharmonic functions
Received by editor(s): March 22, 1996
Additional Notes: The author’s research was supported by GA AV ČR grants C1019601 and 119106 and by GA ČR grant 201/96/0411.
Article copyright: © Copyright 1997 American Mathematical Society