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Characterization of Banach valued BMO functions and UMD Banach spaces by using Bessel convolutions

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Abstract

In this paper we consider the space \({{{BMO}_o(\mathbb{R}, X)}}\) of bounded mean oscillations and odd functions on \({{\mathbb{R}}}\) taking values in a UMD Banach space X. The functions in \({{{BMO}_o(\mathbb{R}, X)}}\) are characterized by Carleson type conditions involving Bessel convolutions and γ-radonifying norms. Also we prove that the UMD Banach spaces are the unique Banach spaces for which certain γ-radonifying Carleson inequalities for Bessel–Poisson integrals of \({{{BMO}_o(\mathbb{R}, X)}}\) functions hold.

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References

  1. Amann H.: Operator-valued Fourier multipliers, vector-valued Besov spaces, and applications. Math. Nachr. 186, 5–56 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  2. Betancor J.J., Buraczewski D., Fariña J.C., Martínez T., Torrea J.L.: Riesz transforms related to Bessel operators. Proc. R. Soc. Edinburgh Sect. A 137, 701– (2007)

    Article  MATH  Google Scholar 

  3. Betancor, J.J., Castro, A.J., Rodríguez-Mesa, L.: Characterization of uniformly convex and smooth Banach spaces by using Carleson measures in Bessel settings. http://arxiv.org/abs/1107.5698 (arXiv:1107.5698v1) (2011, preprint)

  4. Betancor J.J., Chicco Ruiz A., Fariña J.C., Rodríguez-Mesa L.: Maximal operators, Riesz transforms and Littlewood-Paley functions associated with Bessel operators on BMO. J. Math. Anal. Appl 363, 310– (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Betancor J.J., Chicco Ruiz A., Fariña J.C., Rodríguez-Mesa L.: Odd \({{{\rm BMO}(\mathbb{R})})}\) functions and Carleson measures in the Bessel setting. Integr. Equ. Oper. Theory 66, 463– (2010)

    Article  MATH  Google Scholar 

  6. Betancor J.J., Fariña J.C., Martínez T., Torrea J.L.: Riesz transform and g-function associated with Bessel operators and their appropriate Banach spaces. Israel J. Math 157, 259– (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Betancor J.J., Marrero I.: On the topology of the space of Hankel convolution operators. J. Math. Anal. Appl. 201, 994–1001 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  8. Blasco O.: On the dual space of \({{H^{1, \infty}_B}}\). Colloq. Math. 55, 253–259 (1988)

    MathSciNet  MATH  Google Scholar 

  9. Bourgain J.: Some remarks on Banach spaces in which martingale difference sequences are unconditional. Ark. Mat. 21, 163–168 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  10. Burkholder D.L.: A geometrical characterization of Banach spaces in which martingale difference sequences are unconditional. Ann. Probab. 9, 997–1011 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  11. Burkholder, D.L.: A geometric condition that implies the existence of certain singular integrals of Banach-space-valued functions. in Conference on harmonic analysis in honor of Antoni Zygmund, vols. I, II. (Chicago, 1981), Wadsworth Math. Ser., Wadsworth pp. 270–286 (1983)

  12. Burkholder, D.L.: Martingales and Fourier analysis in Banach spaces. in Probability and analysis (Varenna, 1985). Volume 1206 of Lecture Notes in Mathematics. Springer, Berlin, pp. 61–108 (1986)

  13. Coifman R.R., Meyer Y., Stein E.M.: Some new function spaces and their applications to harmonic analysis. J. Funct. Anal. 62, 304–335 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fridli S.: Hardy spaces generated by an integrability condition. J. Approx. Theory 113, 91–109 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  15. Garnett, J.B.: Bounded analytic functions, 1st edn. Volume 236 of Graduate Texts in Mathematics, Springer, New York (2007)

  16. Girardi M., Weis L.: Operator-valued Fourier multiplier theorems on L p (X) and geometry of Banach spaces. J. Funct. Anal. 204, 320–335 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gosselin J., Stempak K.: A weak-type estimate for Fourier-Bessel multipliers. Proc. Am. Math. Soc. 106, 655–662 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities. Cambridge, at the University Press (1934)

  19. Hirschman, I.I. Jr.: Variation diminishing Hankel transforms. J. Anal. Math. 8, 307–336 (1960/1961)

    Google Scholar 

  20. Hytönen T.: Convolutions, multipliers and maximal regularity on vector-valued Hardy spaces. J. Evol. Equ. 5, 205–225 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hytönen T.: Littlewood–Paley–Stein theory for semigroups in UMD spaces. Rev. Mat. Iberoam 23, 973–1009 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hytönen T., van Neerven J., Portal P.: Conical square function estimates in UMD Banach spaces and applications to H ∞-functional calculi. J. Anal. Math. 106, 317–351 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  23. Hytönen T., Weis L.: The Banach space-valued BMO, Carleson’s condition, and paraproducts. J. Fourier Anal. Appl 16, 495–513 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kaiser C., Weis L.: Wavelet transform for functions with values in UMD spaces. Studia Math. 186, 101–126 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kwapień S.: Isomorphic characterizations of inner product spaces by orthogonal series with vector valued coefficients. Studia Math. 44, 583–595 (1972)

    MathSciNet  MATH  Google Scholar 

  26. Lebedev N.N.: Special functions and their applications. Dover Publications Inc., New York (1972)

    MATH  Google Scholar 

  27. Marrero I., Betancor J.J.: Hankel convolution of generalized functions. Rend. Mat. Appl. 15(7), 351–380 (1995)

    MathSciNet  MATH  Google Scholar 

  28. Martínez T., Torrea J.L., Xu Q.: Vector-valued Littlewood–Paley–Stein theory for semigroups. Adv. Math. 203, 430–475 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  29. Muckenhoupt B., Stein E.M.: Classical expansions and their relation to conjugate harmonic functions. Trans. Am. Math. Soc 118, 17–92 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  30. Pisier, G.: Martingales in Banach spaces (in connection with type and cotype). Course IHP (2011)

  31. Rubio de Francia, J.L.: Martingale and integral transforms of Banach space valued functions. In: Probability and Banach spaces (Zaragoza, 1985). Volume 1221 of Lecture Notes in Mathematics. Springer, Berlin pp. 195–222 (1986)

  32. Stein, E.M.: Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Volume 43 of Princeton Mathematical Series, Princeton University Press, Princeton (1993)

  33. van Eijndhoven S.J.L., Graaf J.: Some results on Hankel invariant distribution spaces. Nederl. Akad. Wetensch. Indag. Math. 45, 77–87 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  34. van Neerven, J.: γ-radonifying operators—a survey. In: The AMSI-ANU workshop on spectral theory and harmonic analysis. Volume 44 of Proc. Centre Math. Appl. Austral. Nat. Univ., Canberra, pp. 1–61 (2010)

  35. van Neerven, J.: Stochastic evolution equations. ISEM Lecture Notes (2007/08)

  36. van Neerven J., Weis L.: Stochastic integration of functions with values in a Banach space. Studia Math. 166, 131–170 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  37. Weinstein A.: Discontinuous integrals and generalized potential theory. Trans. Am. Math. Soc. 63, 342–354 (1948)

    Article  MATH  Google Scholar 

  38. Zemanian A.H.: A distributional Hankel transformation. SIAM J. Appl. Math. 14, 561–576 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  39. Zemanian A.H.: Generalized integral transformations. Dover Publications Inc., New York (1987)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Departamento de Análisis Matemático, Universidad de La Laguna, Campus de Anchieta, Avda. Astrofísico Francisco Sánchez, s/n, 38271, La Laguna, Sta. Cruz de Tenerife, Spain

    Jorge J. Betancor, Alejandro J. Castro & Lourdes Rodríguez-Mesa

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  1. Jorge J. Betancor
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  2. Alejandro J. Castro
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  3. Lourdes Rodríguez-Mesa
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Correspondence to Jorge J. Betancor.

Additional information

The authors are partially supported by MTM2010/17974. A. J. Castro is also supported by a FPU grant from the Government of Spain.

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Betancor, J.J., Castro, A.J. & Rodríguez-Mesa, L. Characterization of Banach valued BMO functions and UMD Banach spaces by using Bessel convolutions. Positivity 17, 535–587 (2013). https://doi.org/10.1007/s11117-012-0189-1

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