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Complex Analysis and Operator Theory

, Volume 7, Issue 4, pp 1019–1048 | Cite as

Characterization of UMD Banach Spaces by Imaginary Powers of Hermite and Laguerre Operators

  • Jorge J. Betancor
  • Alejandro J. Castro
  • Jezabel Curbelo
  • Lourdes Rodríguez-MesaEmail author
Article

Abstract

In this paper we characterize the Banach spaces with the UMD property by means of L p -boundedness properties for the imaginary powers of the Hermite and Laguerre operators. In order to do this we need to obtain pointwise representations for the Laplace transform type multipliers associated with Hermite and Laguerre operators.

Keywords

Laguerre operator Laplace transform type multipliers UMD spaces Imaginary powers 

Mathematics Subject Classification (2010)

42C05 (primary) 42C15 (secondary) 

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References

  1. 1.
    Abu-Falahah I., Torrea J.L.: Hermite function expansions versus Hermite polynomial expansions. Glasgow Math. J. 48, 203–215 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Betancor J.J., Castro A.J., Curbelo J.: Spectral multipliers for multidimensional Bessel operators. J. Fourier Anal. Appl. 17(5), 932–975 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Betancor J.J., Fariña J.C., Rodríguez-Mesa L., Sanabria A., Torrea J.L.: Transference between Laguerre and Hermite settings. J. Funct. Anal. 254, 826–850 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Betancor J.J., Martínez M.T., Rodríguez-Mesa L.: Laplace transform type multipliers for Hankel transforms. Can. Math. Bull. 51, 487–496 (2008)zbMATHCrossRefGoogle Scholar
  5. 5.
    Betancor J.J., Molina S.M., Rodríguez-Mesa L.: Area Littlewood-Paley functions associated with Hermite and Laguerre operators. Potent. Anal. 34(4), 345–369 (2011)zbMATHCrossRefGoogle Scholar
  6. 6.
    Bourgain J.: Some remarks on Banach spaces in which martingale difference sequences are unconditional. Ark. Mat. 21, 163–168 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Burkholder, D.L.: A geometric condition that implies the existence of certain singular integrals on Banach-space-valued functions. In: Conference on Harmonic Analysis in honor of Antoni Zygmund, University of Chicago, 1981, Wadswoth Internat. Group, California, vol. 1, pp. 270–286 (1983)Google Scholar
  8. 8.
    Chicco Ruiz A., Harboure E.: Weighted norm inequalities for the heat-diffusion Laguerre’s semigroups. Math. Z. 257, 329–354 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    De Nápoli P.L., Drelichman I., Durán R.G.: Multipliers of Laplace transform type for Laguerre and Hermite expansions. Studia Math. 203, 265–290 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Dore G., Venni A.: On the closedness of the sum of two closed operators. Math. Z. 196(2), 189–201 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    García-Cuerva J., Mauceri G., Sjögren P., Torrea J.L.: Spectral multipliers for the Ornstein–Uhlenbeck semigroup. J. Anal. Math. 78, 281–305 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Guerre-Delabrière S.: Some remarks on complex powers of (−Δ) and UMD spaces. Ill. Math. J. 35, 401–407 (1991)zbMATHGoogle Scholar
  13. 13.
    Gutiérrez C., Incognito A., Torrea J.L.: Riesz-transforms, g-functions and multipliers for the Laguerre semigroup. Houst. J. Math. 27, 579–592 (2001)zbMATHGoogle Scholar
  14. 14.
    Harboure E., Torrea J.L., Viviani B.: Vector-valued extensions of operators related to the Ornstein–Uhlenbeck semigroup. J. d’Analysen 91, 1–29 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Hytönen T.: Aspects of probabilistic Litllewood-Paley theory in Banach spaces, pp. 343–355. Walter de Gruyter, Berlin (2007)Google Scholar
  16. 16.
    Hytönen T.: Littlewood–Paley–Stein theory for semigroups in UMD spaces. Rev. Mat. Iberoam. 23, 973–1009 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Hytönen T., Weis L.: On the necessity of property (α) for some vector-valued multiplier theorems. Arch. Math. (Basel) 90(1), 44–52 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Kunstmann, P.C., Weis, L.: Maximal L p-regularity for parabolic equations, Fourier multiplier theorems and H -functional calculus. In: Functional analytic methods for evolution equations , Lecture Notes in Mathematics, vol. 1855, pp. 65–311. Springer, Berlin (2004)Google Scholar
  19. 19.
    McConnell T.M.: On Fourier multiplier transformations of Banach-valued functions. Trans. Am. Math. Soc. 285, 739–757 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Martínez M.T.: Multipliers of Laplace transform type for ultraspherical expansions. Math. Nachr. 281, 978–988 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Menárguez T., Pérez S., Soria F.: Pointwise and norm estimates for operators associated with the Ornstein–Uhlenbeck semigroup. C. R. Acad. Sci. Paris Sér. I Math. 326(1), 25–30 (1998)zbMATHCrossRefGoogle Scholar
  22. 22.
    Muckenhoupt B.: Poisson integrals for Hermite and Laguerre expansions. Trans. Am. Math. Soc. 139, 231–242 (1969)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Nowak A., Stempak K.: Riesz transforms and conjugacy for Laguerre function expansions of Hermite type. J. Funct. Anal. 244(2), 399–443 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Pisier G.: Some results on Banach spaces without local unconditional structure. Compos. Math. 37(1), 3–19 (1978)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Sasso E.: Spectral multipliers of Laplace transform type for the Laguerre operator. Bull. Aust. Math. Soc. 69, 255–266 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Sjögren, P.: On the maximal function for the Mehler kernel. Harmonic analysis (Cortona, 1982), Lecture Notes in Math., vol. 992, pp. 73–82. Springer, Berlin (1983)Google Scholar
  27. 27.
    Stein E.M.: Topics in harmonic analysis related to the Littlewood–Paley theory. Annals of Mathematics Studies. Princeton Univ. Press, Princeton (1970)Google Scholar
  28. 28.
    Stempak K., Torrea J.L.: Poisson integrals and Riesz transforms for Hermite function expansions with weights. J. Funct. Anal. 202, 443–472 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Szarek, T.: Multipliers of Laplace type transform type in certain Dunkl and Laguerre settings. Preprint (2011). arXiv:1101.4139v1Google Scholar
  30. 30.
    Szegö G.: Orthogonal polynomials, Colloquium Publ., Vol. XXIII. Amer. Math. Soc, Providence (1975)Google Scholar
  31. 31.
    Thangavelu S.: Lectures on Hermite and Laguerre expansions, Mathematical Notes, vol. 42. Princeton Univ. Press, Princeton (1993)Google Scholar
  32. 32.
    Xu Q.: Littlewood–Paley theory for functions with values in uniformly convex spaces. J. Reine Angew. Math. 504, 195–226 (1998)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Wróbel, B.: Laplace type multipliers for Laguerre function expansions of Hermite type. Preprint (2010)Google Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  • Jorge J. Betancor
    • 1
  • Alejandro J. Castro
    • 1
  • Jezabel Curbelo
    • 2
  • Lourdes Rodríguez-Mesa
    • 1
    Email author
  1. 1.Departamento de Análisis MatemáticoUniversidad de la LagunaLa Laguna (Sta. Cruz de Tenerife)Spain
  2. 2.Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Consejo Superior de Investigaciones CientíficasMadridSpain

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