Skip to main content
SpringerLink
Log in

L p–dimension free boundedness for Riesz transforms associated to Hermite functions⋆

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

Riesz transforms associated to Hermite functions were introduced by S. Thangavelu, who proved that they are bounded operators on 1<p< ∞. In this paper we give a different proof that allows us to show that the L p−norms of these operators are bounded by a constant not depending on the dimension d. Moreover, we define Riesz transforms of higher order and free dimensional estimates of the L p−bounds of these operators are obtained. In order to prove the mentioned results we give an extension of the Littlewood-Paley theory that we believe of independent interest.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Coulhon, T., Muller, D., Zienkiewicz, J.: About Riesz transforms on the Heisenberg groups. Math. An. 305, 369–379 (1996)

    MathSciNet  MATH  Google Scholar 

  2. Duoandikoetxea, J., Rubio de Francia, J.L.: Estimations indépendantes de la dimension pour les transformées de Riesz. C. R. Acad. Sci. Paris Sér. I Math. 300, 193–196 (1985)

    MathSciNet  MATH  Google Scholar 

  3. Gosselin, J., Stempak, K.: Conjugate expansions for Hermite functions. Illinois J. Math. 38, 177–197 (1994)

    MathSciNet  MATH  Google Scholar 

  4. Gundy, R.: Some Topics in Probability and Analysis. 70, Conference Board of the Math. Sciences, Amer. Math. Soc., Providence, RI, 1989

  5. Gutiérrez, C.: On the Riesz transforms for Gaussian Measures. J. Funct. Anal. 120, 107–134 (1994)

    Article  MathSciNet  Google Scholar 

  6. Gutiérrez, C., Segovia, C., Torrea, J.L.: On higher Riesz transforms for gaussian measures. J. Fourier Anal. and Appl. 2, 583–596 (1996)

    MathSciNet  Google Scholar 

  7. Krivine, J.L.: Théorèmes de factorisation dans les espaces réticulés. Seminarie Maurey-Schwartz, Exposés École Polytechnique, Paris, 1973-1974, pp. 22–23

  8. Lindenstrauss, J., Tzafriri, L.: Classical Banach spaces II, function spaces. Springer Ser. Modern Surveys Math. 97, Springer-Verlag, Berlin, 1979

  9. Meyer, P.A.: Transformations de Riesz pour les lois Gaussienes. Lecture Notes in Math. 1059, Springer-Verlag, 1984, pp. 179–193

  10. Pisier, G.: Riesz transforms: a simpler analytic proof of P. A. Meyer’s inequality. Lecture Notes in Math., 1321, Springer-Verlag, New York, 1988, pp. 485–501

  11. Rubio de Francia, J.L., Ruiz, F.J., Torrea, J.L.: Calderón-Zygmund theory for operator-valued kernels. Adv. Math. 62, 7–48 (1986)

    MATH  Google Scholar 

  12. Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton Univ. Press, Princeton, NJ, 1970

  13. Stein, E.M.: Topics in Harmonic Analysis related to the Littlewood-Paley Theory. Ann. of Math. Stud. 63, Princeton Univ. Press, Princeton, NJ, 1970

  14. Stein, E.M.: Some results in harmonic analysis in ℝn, for n → ∞. Bull. Am. Math. Soc. (New Series) 9(1), 71–73 (1983)

    MATH  Google Scholar 

  15. Szegö, G.: Orthogonal polynomials. Amer. Math. Soc. Colloq. Publ. 23, Amer. Math. Soc., Providence RI, 1975

  16. Thangavelu, S.: Lectures on Hermite and Laguerre expansions. Math. Notes 42, Princeton Univ. Press, Princeton, 1993

  17. Yosida, K.: Functional Analysis. Springer Verlag, Berlin, 1965

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemática, Facultad de Ingeniería Química, Universidad Nacional del Litoral, 3000, Santa Fe y CONICET, Argentina

    E. Harboure

  2. Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, Pabellón I 1428 ciudad de buenos aires y CONICET, Argentina

    L. de Rosa & C. Segovia

  3. Facultad de Ciencias, Universidad Autónoma de Madrid, Ciudad Universitaria de Canto Blanco, 28049, Madrid, Spain

    J. L. Torrea

Authors
  1. E. Harboure
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. L. de Rosa
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. C. Segovia
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. J. L. Torrea
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. L. Torrea.

Additional information

Mathematical Subject Classification (2000):42B20, 42B25, 42C10

Partially supported by Instituto Argentino de Matemática CONICET, Convenio Universidad Autónoma de Madrid-Universidad Nacional del Litoral, UBACYT 2000-2002 and Ministerio de Ciencia y Tecnologí BFM2002-04013-C02-02

Rights and permissions

Reprints and permissions

About this article

Cite this article

Harboure, E., de Rosa, L., Segovia, C. et al. L p–dimension free boundedness for Riesz transforms associated to Hermite functions⋆ . Math. Ann. 328, 653–682 (2004). https://doi.org/10.1007/s00208-003-0501-2

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-003-0501-2

Keywords

Search

Navigation