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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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 $\Omega $ be a domain in $\mathbb {C}^{n}$, $F$ a nonnegative and $G$ a positive function on $\Omega $ such that $1/G$ is locally bounded, $A^{2}_{\alpha }$ the space of all holomorphic functions on $\Omega $ square-integrable with respect to the measure $F^{\alpha }G\,d\lambda $, where $d\lambda $ is the $2n$-dimensional Lebesgue measure, and $ K_{\alpha }(x,y)$ the reproducing kernel for $A^{2}_{\alpha }$. It has been known for a long time that in some special situations (such as on bounded symmetric domains $\Omega $ with $G=\text {\bf 1}$ and $F=\,$the Bergman kernel function) the formula

\begin{equation*}\lim _{\alpha \to +\infty }K_{\alpha }(x,x)^{1/\alpha }=1/F(x) \tag {$*$} \end{equation*}

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 $-\log F$ can be approximated by certain pluriharmonic functions lying below it. For instance, ($*$) holds if $-\log F$ 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 $K_{\alpha }(x,y)^{1/\alpha }$ for $x\neq y$, 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 $K_{\alpha }(x,y)$ must have zeroes for all $\alpha $ sufficiently large.


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Additional Information

Miroslav Englis
Affiliation: Mathematical Institute of the Academy of Sciences, Žitná 25, 11567 Prague 1, Czech Republic
Email: englis@math.cas.cz

DOI: http://dx.doi.org/10.1090/S0002-9947-97-01843-6
PII: S 0002-9947(97)01843-6
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