## Asymptotic behaviour of reproducing kernels of weighted Bergman spaces

HTML articles powered by AMS MathViewer

- by Miroslav Engliš PDF
- Trans. Amer. Math. Soc.
**349**(1997), 3717-3735 Request permission

## 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 $G=\mathbf {1}$ 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.## References

- V. Bargmann,
*On a Hilbert space of analytic functions and an associated integral transform*, Comm. Pure Appl. Math.**14**(1961), 187–214. MR**157250**, DOI 10.1002/cpa.3160140303 - Heinz Bauer,
*Approximation and abstract boundaries*, Amer. Math. Monthly**85**(1978), no. 8, 632–647. MR**508225**, DOI 10.2307/2320332 - F. A. Berezin,
*Quantization*, Izv. Akad. Nauk SSSR Ser. Mat.**38**(1974), 1116–1175 (Russian). MR**0395610** - C. A. Berger and L. A. Coburn,
*Toeplitz operators and quantum mechanics*, J. Funct. Anal.**68**(1986), no. 3, 273–299. MR**859136**, DOI 10.1016/0022-1236(86)90099-6 - C. A. Berger and L. A. Coburn,
*Toeplitz operators on the Segal-Bargmann space*, Trans. Amer. Math. Soc.**301**(1987), no. 2, 813–829. MR**882716**, DOI 10.1090/S0002-9947-1987-0882716-4 - Stefan Bergman,
*The kernel function and conformal mapping*, Second, revised edition, Mathematical Surveys, No. V, American Mathematical Society, Providence, R.I., 1970. MR**0507701** - Morgan Ward,
*Ring homomorphisms which are also lattice homomorphisms*, Amer. J. Math.**61**(1939), 783–787. MR**10**, DOI 10.2307/2371336 - J. Rawnsley, M. Cahen, and S. Gutt,
*Quantization of Kähler manifolds. I. Geometric interpretation of Berezin’s quantization*, J. Geom. Phys.**7**(1990), no. 1, 45–62. MR**1094730**, DOI 10.1016/0393-0440(90)90019-Y - Miroslav Engliš,
*Asymptotics of reproducing kernels on a plane domain*, Proc. Amer. Math. Soc.**123**(1995), no. 10, 3157–3160. MR**1277107**, DOI 10.1090/S0002-9939-1995-1277107-0 - Miroslav Engliš,
*Asymptotics of the Berezin transform and quantization on planar domains*, Duke Math. J.**79**(1995), no. 1, 57–76. MR**1340294**, DOI 10.1215/S0012-7094-95-07902-2 - Miroslav Engliš,
*Berezin quantization and reproducing kernels on complex domains*, Trans. Amer. Math. Soc.**348**(1996), no. 2, 411–479. MR**1340173**, DOI 10.1090/S0002-9947-96-01551-6 - J. Faraut and A. Korányi,
*Function spaces and reproducing kernels on bounded symmetric domains*, J. Funct. Anal.**88**(1990), no. 1, 64–89. MR**1033914**, DOI 10.1016/0022-1236(90)90119-6 - J. Faraut, A. Korányi,
*Analysis on symmetric cones*, Clarendon Press, Oxford, 1994. - M. V. Fedoryuk,
*Asimptotika: integraly i ryady*, Spravochnaya Matematicheskaya Biblioteka. [Mathematical Reference Library], “Nauka”, Moscow, 1987 (Russian). MR**950167** - Sigurđur Helgason,
*Differential geometry and symmetric spaces*, Pure and Applied Mathematics, Vol. XII, Academic Press, New York-London, 1962. MR**0145455** - L. K. Hua,
*Harmonic analysis of functions of several complex variables in the classical domains*, American Mathematical Society, Providence, R.I., 1963. Translated from the Russian by Leo Ebner and Adam Korányi. MR**0171936** - Svante Janson, Jaak Peetre, and Richard Rochberg,
*Hankel forms and the Fock space*, Rev. Mat. Iberoamericana**3**(1987), no. 1, 61–138. MR**1008445**, DOI 10.4171/RMI/46 - Maciej Klimek,
*Pluripotential theory*, London Mathematical Society Monographs. New Series, vol. 6, The Clarendon Press, Oxford University Press, New York, 1991. Oxford Science Publications. MR**1150978** - Irwin Kra,
*Automorphic forms and Kleinian groups*, Mathematics Lecture Note Series, W. A. Benjamin, Inc., Reading, Mass., 1972. MR**0357775** - O. Loos,
*Bounded symmetric domains and Jordan pairs*, University of California, Irvine, 1977. - I. Netuka, J. Veselý,
*On harmonic functions. Solution to problem 6393 [1982, 502] proposed by G.A. Edgar*, Amer. Math. Monthly**91**(1984), 61–62. - Walter Rudin,
*Function theory in the unit ball of $\textbf {C}^{n}$*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 241, Springer-Verlag, New York-Berlin, 1980. MR**601594** - Maciej Skwarczyński,
*Biholomorphic invariants related to the Bergman function*, Dissertationes Math. (Rozprawy Mat.)**173**(1980), 59. MR**575756** - E.C. Titchmarsh,
*The theory of functions*, Oxford University Press, Oxford, 1939. - A. Unterberger and H. Upmeier,
*The Berezin transform and invariant differential operators*, Comm. Math. Phys.**164**(1994), no. 3, 563–597. MR**1291245** - W. B. Arveson, A. S. Mishchenko, M. Putinar, M. A. Rieffel, and Ş. Strătilă (eds.),
*Operator algebras and topology*, Pitman Research Notes in Mathematics Series, vol. 270, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1992. MR**1189176** - Zoran Vondraček,
*On some extremal elements in the cone $\scr H^\textrm {inf}$*, Glas. Mat. Ser. III**27(47)**(1992), no. 2, 241–249 (English, with English and Serbo-Croatian summaries). MR**1244641** - Zoran Vondraček,
*The Martin kernel and infima of positive harmonic functions*, Trans. Amer. Math. Soc.**335**(1993), no. 2, 547–557. MR**1104202**, DOI 10.1090/S0002-9947-1993-1104202-1 - H. Wu,
*Old and new invariant metrics on complex manifolds*, Several complex variables (Stockholm, 1987/1988) Math. Notes, vol. 38, Princeton Univ. Press, Princeton, NJ, 1993, pp. 640–682. MR**1207887**

## Additional Information

**Miroslav Engliš**- Affiliation: Mathematical Institute of the Academy of Sciences, Žitná 25, 11567 Prague 1, Czech Republic
- Email: englis@math.cas.cz
- 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.
- © Copyright 1997 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**349**(1997), 3717-3735 - MSC (1991): Primary 30C40, 32H10; Secondary 31C10, 30E15
- DOI: https://doi.org/10.1090/S0002-9947-97-01843-6
- MathSciNet review: 1401769