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On Boundedness of Calderón–Toeplitz Operators

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Abstract

We study the boundedness of Toeplitz-type operators defined in the context of the Calderón reproducing formula considering the specific wavelets whose Fourier transforms are related to Laguerre polynomials. Some sufficient conditions for simultaneous boundedness of these Calderón–Toeplitz operators on each wavelet subspace for unbounded symbols are given, where investigating the behavior of certain sequence of iterated integrals of symbols is helpful. A number of examples and counterexamples is given.

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References

  1. Abreu, L.D.: Super-wavelets versus poly-Bergman spaces. Preprint (2009). arXiv:0909.4830v1

  2. Calderón A.: Intermediate spaces and interpolation, the complex method. Studia Math. 24, 113–190 (1964)

    MathSciNet  MATH  Google Scholar 

  3. Daubechies I.: Ten Lectures on Wavelets. SIAM, Philadelphia (1992)

    MATH  Google Scholar 

  4. Gradshteyn I.S., Ryzhik I.M.: Tables of Integrals, Series and Products, 7th edn. Academic Press/Elsevier, San Diego (2007)

    Google Scholar 

  5. Grudsky S., Karapetyans A., Vasilevski N.: Dynamics of properties of Toeplitz operators on the upper half-plane: parabolic case. J. Oper. Theory 52, 1–31 (2004)

    Google Scholar 

  6. Hutník O.: On the structure of the space of wavelet transforms. C. R. Acad. Sci. Paris Ser. I 346, 649–652 (2008)

    MATH  Google Scholar 

  7. Hutník O.: On Toeplitz-type operators related to wavelets. Integral Equ. Oper. Theory 63(1), 29–46 (2009)

    Article  MATH  Google Scholar 

  8. Hutník O.: A note on wavelet subspaces. Monatsh. Math. 160(1), 59–72 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hutník, O.: Wavelets from Laguerre polynomials and Toeplitz-type operators. Preprint (2011). arXiv:1102.3707

  10. Nowak K.: On Calderón-Toeplitz operators. Monatsh. Math. 116, 49–72 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  11. Nowak K.: Local Toeplitz operators based on wavelets: phase space patterns for rough wavelets. Studia Math. 119, 37–64 (1996)

    MathSciNet  MATH  Google Scholar 

  12. Quiroga-Barranco R., Vasilevski N.L.: Commutative C *-algebras of Toeplitz operators on the unit ball, I. Bargmann-type transforms and spectral representations of Toeplitz operators. Integral Equ. Oper. Theory 59(3), 379–419 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Rainville E.D.: Special Functions. Macmillan, New York (1960)

    MATH  Google Scholar 

  14. Rochberg, R.: Toeplitz and Hankel operators, wavelets, NWO sequences, and almost diagonalization of operators. In: Arveson, W.B., Douglas, R.G. (eds.) Proceedings of Symposia in Pure Mathematics, part I, vol. 51, pp. 425–444. American Mathematical Society, Providence (1990)

  15. Rochberg, R.: Eigenvalue estimates for Calderón-Toeplitz operators. In: Jarosz, K. (ed.) Lecture Notes in Pure and Applied Mathematics, vol. 136, pp. 345–357. Dekker, New York (1992)

  16. Vasilevski N.L.: On the structure of Bergman and poly-Bergman spaces. Integral Equ. Oper. Theory 33, 471–488 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  17. Vasilevski N.L.: Commutative Algebras of Toeplitz Operators on the Bergman Space. Series: Operator Theory: Advances and Applications, vol. 185. Birkhäuser, Basel (2008)

    Google Scholar 

  18. Wang Z.X., Guo D.R.: Special Functions. World Scientific, Singapore (1989)

    Google Scholar 

  19. Wong M.W.: Wavelet Transforms and Localization Operators. Series: Operator Theory: Advances and Applications, vol. 136. Birkhäuser, Basel (2002)

    Google Scholar 

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Authors and Affiliations

  1. Departamento de Matemáticas, CINVESTAV del IPN, Apartado Postal 14-740, 07000, Mexico, D.F., Mexico

    Ondrej Hutník

  2. Institute of Mathematics, Faculty of Science, Pavol Jozef Šafárik University, Jesenná 5, 040 01, Košice, Slovakia

    Ondrej Hutník

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  1. Ondrej Hutník
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Correspondence to Ondrej Hutník.

Additional information

This research was partially supported by Grant VVGS 45/10-11.

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Hutník, O. On Boundedness of Calderón–Toeplitz Operators. Integr. Equ. Oper. Theory 70, 583–600 (2011). https://doi.org/10.1007/s00020-011-1883-2

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  • DOI: https://doi.org/10.1007/s00020-011-1883-2

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