Skip to main content
SpringerLink
Log in

A Transference Result of the L p Continuity from Jacobi Riesz Transform to the Gaussian and Laguerre Riesz Transforms

  • Published:
Journal of Fourier Analysis and Applications Aims and scope Submit manuscript

Abstract

In this paper we develop a transference method to obtain the Gaussian-Riesz transform’s L p-continuity and the Laguerre-Riesz transform’s L p-continuity from the L p-continuity of the Jacobi-Riesz transform, in dimension one, using the well known asymptotic relations between Jacobi polynomials and Hermite and Laguerre polynomials.

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. Andrews, G.E., Askey, R., Roy, R.: Special Functions. Enciclopedia of Mathematics and Its Applications, vol. 7. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  2. Bayley, W.N.: Generalized Hypergeometric Series. Cambridge Tracs in Mathematics and Mathematical Physics, vol. 32. Cambridge University Press, New York (1964)

    Google Scholar 

  3. Betancor, J., Fariña, J., Rodriguez, L., Sanabria, A.: Transferring boundedness from conjugate operators associated whit Jacobi, Laguerre and Fourier-Bessel expansions to conjugate operators in the Hankel setting. J. Fourier Anal. Appl. 14(4), 493–513 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cafarelli, L.: Sobre la conjugación y sumabilidad de series de Jacobi. Ph.D. Thesis, Universidad de Buenos Aires, Argentina (1971)

  5. Gasper, G.: Unpublished manuscrip

  6. Gundy, R.: Sur les transformations de Riesz pour le semigroup d’Ornstein-Uhlenbeck. C.R. Acad. Sci. 303 (série I), 967–970 (1986)

    MathSciNet  Google Scholar 

  7. Gutiérrez, C.: On the Riesz transforms for the Gaussian measure. J. Funct. Anal. 120(1), 107–134 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  8. Li, Zh.-K.: Conjugated Jacobi series and conjugated functions. J. Approx. Theory 86, 179–196 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  9. Meyer, P.A.: Transformations de Riesz pour les lois Gaussiennes. In: Lectures Notes in Math., vol. 1059, pp. 179–193. Springer, Berlin (1984)

    Google Scholar 

  10. Muckenhoupt, B.: Hermite conjugate expansions. Trans. Am. Math. Soc. 139, 243–260 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  11. Muckenhoupt, B.: Conjugate functions for Laguerre expansions. Trans. Am. Math. Soc. 147, 403–418 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  12. Muckenhoupt, B., Stein, E.M.: Classical expansions. Trans. Am. Math. Soc. 147, 17–92 (1965)

    Article  MathSciNet  Google Scholar 

  13. Nowak, A.: On Riesz transforms for Laguerre expansions. J. Funct. Anal. 215(1), 217–240 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. Nowak, A., Sjögren, P.: Riesz transforms for Jacobi expansions. J. Anal. Math. 104, 341–369 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Nowak, A., Stempak, K.: L 2-theory of Riesz transforms for orthogonal expansions. J. Fourier Anal. Appl. 12(6), 675–711 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Pérez, S., Soria, F.: Operators associated with the Ornstein-Uhlenbeck semigroup. J. Lond. Math. Soc. (2) 61(3), 857–871 (2000)

    Article  MATH  Google Scholar 

  17. Pisier, G.: Riesz transform: a simpler analytic proof of P.A. Meyer inequality. In: Lectures Notes in Math, vol. 1321, pp. 485–501. Springer, Berlin (1988)

    Google Scholar 

  18. Stein, E.M.: Singular Integrals and Differenciability Properties of Functions. Princeton Univ. Press, Princeton (1970)

    Google Scholar 

  19. Stein, E.M.: Topics in Harmonic Analysis Related to the Littlewood-Paley Theory. Princeton Univ. Press, Princeton (1970)

    MATH  Google Scholar 

  20. Szegö, G.: Orthogonal Polynomials. Colloq. Publ., vol. 23. Am. Math. Soc., Providence (1959)

    MATH  Google Scholar 

  21. Urbina, W.: Singular Integrals with respect to the Gaussian measure. Scuola Normale Superiore di Pisa. Classe di Science. Serie IV, vol. XVIII, 4, pp. 531–567 (1990)

  22. Urbina, W.: Operators semigroups associated to classical orthogonal polynomials and functional inequalities. Seminaires et Congres. French Mathematical Society (SMF) 200812

  23. Zygmund, A.: Trigonometric Series, 2nd. edn. Cambridge Univ. Press, Cambridge (1959)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departmento de Matemáticas, Universidad Nacional Experimental Francisco de Miranda, Punto Fijo, Venezuela

    Eduard Navas

  2. Department of Mathematics and Actuarial Sciences, Roosevelt University, Chicago, IL, 60605, USA

    Wilfredo O. Urbina

Authors
  1. Eduard Navas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wilfredo O. Urbina
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wilfredo O. Urbina.

Additional information

Communicated by Fernando Soria.

Dedicated to professor Calixto P. Calderón with deep admiration.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Navas, E., Urbina, W.O. A Transference Result of the L p Continuity from Jacobi Riesz Transform to the Gaussian and Laguerre Riesz Transforms. J Fourier Anal Appl 19, 910–942 (2013). https://doi.org/10.1007/s00041-013-9284-3

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00041-013-9284-3

Keywords

Mathematics Subject Classification

Search

Navigation