Weighted Bergman kernel function, admissible weights and the Ramadanov theorem

P. Wojcicki

The Bergman kernel is an important tool in geometric function theory, both in one and several complex variables. It turned out that not only \regular" Bergman kernel, but also weighted one can be useful, particularly from quantum theory point of view ([3], [7]). But in general, it is difficult to say anything about the kernel of a given domain. One of the classic results is Ramadanov's theorem. Theorem. Let Let $\Omega_1\Subset\Omega_2\Subset\Omega_3\ldots$ be an increasing sequence of domains and set $\Omega\equiv\bigcup_j\Omega_j$. Then, {for all $j$,} $K_{\Omega_j}\rightarrow K_\Omega$ uniformly on compact subsets of $\Omega\times\Omega.$ Some versions of a weighted analog of this theorem are known ([4, Pr. 3.17; Th. 3.18] for instance), but considered weights are of a special form, as a modulus of holomorphic functions or $C^2$ functions, or as a product of one of those by a given weight $\psi$. In the present paper we provide a weighted analog of this theorem, for very general weights.

1. S. Bergman, The kernel function and conformal mapping, A.M.S. Survey Number V, 2nd Edition, 1970.

2. H.P. Boas, Counterexample to the Lu Qi-Keng conjecture, Proc. Amer. Math. Soc., 97 (1986), №2, 374–375.

3. M. Englis, Weighted Bergman kernels and quantization, Comm. Math. Phys., 227 (2002), №2, 211–241.

4. R.L. Jacobson, Weighted Bergman Kernel Functions and the Lu Qi-Keng problem, Thesis (Ph.D.) – Texas A&M University, 2012.

5. S.G. Krantz, A new proof and a generalization of Ramadanov’s theorem, Complex variables and elliptic equations, 51 (2006), №12, 1125–1128.

6. S.G. Krantz, Geometric analysis of the Bergman kernel and metric, New York, Springer, 2013.

7. A. Odzijewicz, On reproducing kernels and quantization of states, Commun. Math. Phys., 114 (1988), 577–597.

8. Z. Pasternak-Winiarski, On the dependence of the reproducing kernel on the weight of integration, Journal of Functional Analysis, 94 (1990), №1, 110–134.

9. Z. Pasternak-Winiarski, On weights which admit the reproducing kernel of Bergman type, Internat. J. Math & Math. Sci., 15 (1992), №1, 1–14.

10. B.V. Shabat, Introduction to Complex Analysis, Part II : Functions of several variables, Translations of Mathematical Monographs, V.110, A.M.S.

11. M. Skwarczynski, T. Mazur, Wstepne twierdzenia teorii funkcji wielu zmiennych zespolonych, Wyd. Krzysztof Biesaga, Warszawa, 2001.

12. Skwarczyґnski, M., Biholomorphic invariants related to the Bergman function, Dissertationes Mathematicae (Rozprawy Matematyczne), V.173, Warsaw, Polish Scientific Publishing Company, 1980.

	Weighted Bergman kernel function, admissible weights and the Ramadanov theorem
Author	P. Wojcicki p.wojcicki@mini.pw.edu.pl Faculty of Mathematics and Information Science, Warsaw University of Technology
Abstract	The Bergman kernel is an important tool in geometric function theory, both in one and several complex variables. It turned out that not only \regular" Bergman kernel, but also weighted one can be useful, particularly from quantum theory point of view ([3], [7]). But in general, it is difficult to say anything about the kernel of a given domain. One of the classic results is Ramadanov's theorem. Theorem. Let Let $\Omega_1\Subset\Omega_2\Subset\Omega_3\ldots$ be an increasing sequence of domains and set $\Omega\equiv\bigcup_j\Omega_j$. Then, {for all $j$,} $K_{\Omega_j}\rightarrow K_\Omega$ uniformly on compact subsets of $\Omega\times\Omega.$ Some versions of a weighted analog of this theorem are known ([4, Pr. 3.17; Th. 3.18] for instance), but considered weights are of a special form, as a modulus of holomorphic functions or $C^2$ functions, or as a product of one of those by a given weight $\psi$. In the present paper we provide a weighted analog of this theorem, for very general weights.
Keywords	weighted Bergman kernel; Bergman space; Ramadanov theorem
Reference	1. S. Bergman, The kernel function and conformal mapping, A.M.S. Survey Number V, 2nd Edition, 1970. 2. H.P. Boas, Counterexample to the Lu Qi-Keng conjecture, Proc. Amer. Math. Soc., 97 (1986), №2, 374–375. 3. M. Englis, Weighted Bergman kernels and quantization, Comm. Math. Phys., 227 (2002), №2, 211–241. 4. R.L. Jacobson, Weighted Bergman Kernel Functions and the Lu Qi-Keng problem, Thesis (Ph.D.) – Texas A&M University, 2012. 5. S.G. Krantz, A new proof and a generalization of Ramadanov’s theorem, Complex variables and elliptic equations, 51 (2006), №12, 1125–1128. 6. S.G. Krantz, Geometric analysis of the Bergman kernel and metric, New York, Springer, 2013. 7. A. Odzijewicz, On reproducing kernels and quantization of states, Commun. Math. Phys., 114 (1988), 577–597. 8. Z. Pasternak-Winiarski, On the dependence of the reproducing kernel on the weight of integration, Journal of Functional Analysis, 94 (1990), №1, 110–134. 9. Z. Pasternak-Winiarski, On weights which admit the reproducing kernel of Bergman type, Internat. J. Math & Math. Sci., 15 (1992), №1, 1–14. 10. B.V. Shabat, Introduction to Complex Analysis, Part II : Functions of several variables, Translations of Mathematical Monographs, V.110, A.M.S. 11. M. Skwarczynski, T. Mazur, Wstepne twierdzenia teorii funkcji wielu zmiennych zespolonych, Wyd. Krzysztof Biesaga, Warszawa, 2001. 12. Skwarczyґnski, M., Biholomorphic invariants related to the Bergman function, Dissertationes Mathematicae (Rozprawy Matematyczne), V.173, Warsaw, Polish Scientific Publishing Company, 1980.
Pages	160-164
Volume	42
Issue	2
Year	2014
Journal	Matematychni Studii
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