Directional derivative estimates for Berezin’s operator calculus

Coburn, L.; Li, Bo

doi:10.1090/S0002-9939-07-09081-8

Directional derivative estimates for Berezin’s operator calculus
HTML articles powered by AMS MathViewer

by L. A. Coburn and Bo Li PDF

Proc. Amer. Math. Soc. 136 (2008), 641-649 Request permission

Abstract:

Directional derivative estimates for Berezin symbols of bounded operators on Bergman spaces of arbitrary bounded domains $\Omega$ in $\mathbb C^n$ are obtained. These estimates also hold in the setting of the Segal-Bargmann space on $\mathbb C^n$. It is also shown that our estimates are sharp at every point of $\Omega$ by exhibiting the optimizers explicitly.

References

F. A. Berezin, Covariant and contravariant symbols of operators, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 1134–1167 (Russian). MR 0350504
F. A. Berezin, Quantization, Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 1116–1175 (Russian). MR 0395610
L. A. Coburn, A Lipschitz estimate for Berezin’s operator calculus, Proc. Amer. Math. Soc. 133 (2005), no. 1, 127–131. MR 2085161, DOI 10.1090/S0002-9939-04-07476-3
L. A. Coburn, Sharp Berezin Lipschitz estimates, Proc. Amer. Math. Soc. 135 (2007), no. 4, 1163–1168. MR 2262921, DOI 10.1090/S0002-9939-06-08569-8
Miroslav Engliš and Genkai Zhang, On the derivatives of the Berezin transform, Proc. Amer. Math. Soc. 134 (2006), no. 8, 2285–2294. MR 2213701, DOI 10.1090/S0002-9939-06-08238-4
I. C. Gohberg and M. G. Kreĭn, Introduction to the theory of linear nonselfadjoint operators, Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, R.I., 1969. Translated from the Russian by A. Feinstein. MR 0246142
Kyong T. Hahn, On completeness of the Bergman metric and its subordinate metric, Proc. Nat. Acad. Sci. U.S.A. 73 (1976), no. 12, 4294. MR 417459, DOI 10.1073/pnas.73.12.4294
Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 514561
Gregor Herbort, Über die Geodätischen der Bergmanmetrik, Schriftenreihe des Mathematischen Instituts der Universität Münster, Ser. 2 [Series of the Mathematical Institute of the University of Münster, Ser. 2], vol. 26, Universität Münster, Mathematisches Institut, Münster, 1983 (German). MR 697235
Marek Jarnicki and Peter Pflug, Invariant distances and metrics in complex analysis, De Gruyter Expositions in Mathematics, vol. 9, Walter de Gruyter & Co., Berlin, 1993. MR 1242120, DOI 10.1515/9783110870312
Bo Li, The Berezin transform and Laplace-Beltrami operator, J. Math. Anal. Appl. 327 (2007), no. 2, 1155–1166. MR 2279995, DOI 10.1016/j.jmaa.2006.04.068
T. Mazur, P. Pflug, and M. Skwarczyński, Invariant distances related to the Bergman function, Proc. Amer. Math. Soc. 94 (1985), no. 1, 72–76. MR 781059, DOI 10.1090/S0002-9939-1985-0781059-0
Takashi Sakai, Riemannian geometry, Translations of Mathematical Monographs, vol. 149, American Mathematical Society, Providence, RI, 1996. Translated from the 1992 Japanese original by the author. MR 1390760, DOI 10.1090/mmono/149
Richard M. Timoney, Bloch functions in several complex variables. I, Bull. London Math. Soc. 12 (1980), no. 4, 241–267. MR 576974, DOI 10.1112/blms/12.4.241