Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-22T05:18:40.304Z Has data issue: false hasContentIssue false
  • English
  • Français

Toeplitz operators on Bergman spaces of polygonal domains

Part of: Special classes of linear operators

Published online by Cambridge University Press:  26 June 2019

Paula Mannersalo
Paula Mannersalo*
Affiliation:
Department of Mathematics and Statistics, P.O. Box 68, FI-00014 University of Helsinki, Finland ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the boundedness of Toeplitz operators with locally integrable symbols on Bergman spaces Ap(Ω), 1 < p < ∞, where Ω ⊂ ℂ is a bounded simply connected domain with polygonal boundary. We give sufficient conditions for the boundedness of generalized Toeplitz operators in terms of ‘averages’ of symbol over certain Cartesian squares. We use the Whitney decomposition of Ω in the proof. We also give examples of bounded Toeplitz operators on Ap(Ω) in the case where polygon Ω has such a large corner that the Bergman projection is unbounded.

Keywords

Toeplitz operatorBergman spaceboundednesspolygonal domainlocally integrable symbolWhitney decompositionSchwarz–Christoffel formula

MSC classification

Primary: 47B35: Toeplitz operators, Hankel operators, Wiener-Hopf operators
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 4 , November 2019 , pp. 1115 - 1136
Copyright
Copyright © Edinburgh Mathematical Society 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Békollé, D., Inégalités à poids pour le projecteur de Bergman dans la boule unité de ℂn, Studia Math 71(3) (1981/1982), 305–323.Google Scholar
2.Békollé, D., Projections sur des espaces de fonctions holomorphes dans des domains planes, Canad. J. Math. 38(1) (1986), 127–157.Google Scholar
3.Duren, P. and Schuster, A., Bergman spaces, Mathematical Surveys and Monographs, Volume 100 (American Mathematical Society, Providence, RI, 2004).Google Scholar
4.Hedenmalm, H., The dual of a Bergman space on simply connected domains, J. Anal. Math. 88(1) (2002), 311–335.Google Scholar
5.Hutson, V. and Pym, J. S., Applications of functional analysis and operator theory, Mathematics in Science and Engineering, Volume 146 (Academic Press, London, 1980).Google Scholar
6.Lanzani, L. and Stein, E. M., Szegö and Bergman projections on non-smooth planar domains, J. Geom. Anal. 14(1) (2004), 63–86.Google Scholar
7.Mannersalo, P., Toeplitz operators with locally integrable symbols on Bergman spaces of bounded simply connected domains, Complex Var. Elliptic Equ. 61(6) (2016), 854–874.Google Scholar
8.Nehari, Z., Conformal mapping (McGraw-Hill, New York, 1952).Google Scholar
9.Peláez, J.Á. and Rättyä, J., Two weight inequality for Bergman projection, J. Math. Pures Appl. 105(1) (2016), 102–130.Google Scholar
10.Pommerenke, C., Boundary behaviour of conformal maps (Springer-Verlag, Berlin, 1992).Google Scholar
11.Shikhvatov, A. M., Spaces of analytic functions in a region with an angle, Math. Notes 18 (1975), 833–839.Google Scholar
12.Solov'ev, A. A., L p-estimates of integral operators associated with spaces of analytic and harmonic functions, Soviet Math. Dokl. 19 (1978), 764–768.Google Scholar
13.Solov'ev, A. A., On the continuity of an integral operator with Bergman kernel in the L p space (Russian), Vest. Leningrad Universiteta 19 (1978), 77–80.Google Scholar
14.Stein, E. M., Singular integrals and differentiability properties of functions (Princeton University Press, Princeton, NJ, 1970).Google Scholar
15.Taskinen, J., Regulated domains and Bergman type projections, Ann. Acad. Sci. Fenn. Math. 28(1) (2003), 55–68.Google Scholar
16.Taskinen, J., Note on the paper ‘Regulated domains and Bergman type projections’, J. Funct. Spaces Appl. 2(2) (2004), 97–106.Google Scholar
17.Taskinen, J. and Virtanen, J., Toeplitz operators on Bergman spaces with locally integrable symbols, Rev. Mat. Iberoam. 26(2) (2010), 693–706.Google Scholar
18.Taskinen, J. and Virtanen, J., On generalized Toeplitz and little Hankel operators on Bergman spaces, Arch. Math. 110(2) (2018), 155–166.Google Scholar
19.Zhu, K., Operator theory in function spaces, Mathematical Surveys and Monographs, 7th edn, Volume 138 (American Mathematical Society, Providence, RI, 2007).Google Scholar