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

  • Miroslav Engliš
Part of the International Society for Analysis, Applications and Computation book series (ISAA, volume 3)


Let Ω be a domain in C n , F and G positive measurable functions on Ω such that 1/F and 1/G are locally bounded, A α 2 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 α 2 (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 ,(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.


Bergman Space Planar Domain Bergman Kernel Hankel Operator Weighted Bergman Space 
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  1. [1]
    P. Ahern, M. Flores, W. Rudin. An invariant volume-mean-value property. J. Funct. Anal., 111, pp. 380–397, 1993.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    J. Arazy, G. Zhang. LQ-estimates of spherical functions and an invariant mean-value property. Integr. Eq. Oper. Theory, 23, pp. 123–144, 1995.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    M. Beals, C. Fefferman, R. Grossman. Strictly pseudoconvex domains in Cn. Bull. Amer. Math. Soc., 8, pp. 125–326, 1983.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    F.A. Berezin. Quantization. Math. USSR Izvestiya, 8, pp. 1109–1163, 1974.CrossRefGoogle Scholar
  5. [5]
    M. Englis`. Asymptotics of the Berezin transform and quantization on planar domains. Duke Math. J., 79, pp. 57–76, 1995.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    M. Englis. Berezin quantization and reproducing kernels on complex domains. Trans. Amer. Math. Soc., 348, pp. 411–479, 1996.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    M. Englis. Asymptotic behavior of reproducing kernels of weighted Bergman spaces. Trans. Amer. Math. Soc., 349, pp. 3717–3735, 1997.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    M. Englis. A mean-value theorem on bounded symmetric domains. Proc. Amer. Math. Soc., to appear.Google Scholar
  9. [9]
    M. Engli. A calculation of asymptotics of the Laplace integral on a Kähler manifold. In preparation.Google Scholar
  10. [10]
    M. Englis, J. Peetre. On the correspondence principle for the quantized annulus. Math. Scand., 78, pp. 183–206, 1996.MathSciNetzbMATHGoogle Scholar
  11. [11]
    H. Fürstenberg. Poisson formula for semisimple Lie groups. Ann. Math., 77, pp. 335–386, 1963.CrossRefGoogle Scholar
  12. [12]
    R. Godement. Une généralisation des représentations de la moyenne pour les fonctions harmoniques. C. R. Acad. Sci. Paris, 234, pp. 2137–2139, 1952.MathSciNetzbMATHGoogle Scholar
  13. [13]
    M. Jarnicki, P. Pflug. Invariant distances and metrics in complex analysis. Walter de Gruyter, 1993.Google Scholar
  14. [14]
    P. Lin, R. Rochberg. Hankel operators on the weighted Bergman spaces with exponential type weights. Integral Equations Operator Theory, 21, pp. 460–483, 1995.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    J. Peetre. The Berezin transform and Ha-plitz operators. J. Operator Theory, 24, pp. 165–186, 1990.MathSciNetzbMATHGoogle Scholar

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© Springer Science+Business Media Dordrecht 1999

Authors and Affiliations

  • Miroslav Engliš
    • 1
  1. 1.Mathematical Institute of the Academy of SciencesPrague 1Czech Republic

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