Improving Bounds for Singular Operators via Sharp Reverse Höolder Inequality for \( A_\infty \)

  • Carmen Ortiz-Caraballo
  • Carlos Pérez
  • Ezequiel Rela
Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 229)

Abstract

In this expository article we collect and discuss some recent results on different consequences of a Sharp Reverse Hölder Inequality for \( A\infty \) weights. For two given operators T and S, we study \( L^{p}(w) \) bounds of Coifman– Fefferman type:
$$ \parallel T\;f \parallel_{L^p}(w)\;\leq \; c_{n,w,p}\parallel S\;f \parallel _{L^p}(w),$$
that can be understood as a way to control T by S.
We will focus on a quantitative analysis of the constants involved and show that we can improve classical results regarding the dependence on the weight w in terms of Wilson’s \( A\infty \) constant
$$ [w]A_{\infty}\; := \; {\rm sup_Q}\frac{1}{w(Q)}\int_{Q}{M({w_\mathcal{X}}_Q)} .$$
We will also exhibit recent improvements on the problem of finding sharp constants for weighted norm inequalities involving several singular operators. In the same spirit as in [10], we obtain mixed \( A_{1}-A_{\infty}\) estimates for the commutator [b,T] and for its higher–order analogue \( T^{k}_{b}\) . A common ingredient in the proofs presented here is a recent improvement of the Reverse Hölder Inequality for \( A_{\infty}\) weights involving Wilson’s constant from [10].

Keywords.

Weighted norm inequalities Reverse Hölder Inequality maximal operators singular integrals Calderón-Zygmund theory commutators. 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    J. Alvarez and C. Pérez, Estimates with A∞ weights for various singular integral operators, Bollettino U.M.I. 7 8-A (1994), 123–133.Google Scholar
  2. [2]
    R.J. Bagby and D.S. Kurtz, Covering lemmas and the sharp function, Proc. Amer. Math. Soc. 93 (1985), 291–296.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    R.J. Bagby and D.S. Kurtz, A rearranged good-λ inequality, Trans. Amer. Math. Soc. 293 (1986), 71–81.MathSciNetMATHGoogle Scholar
  4. [4]
    S.M. Buckley, Estimates for operator norms on weighted spaces and reverse Jensen inequalities, Trans. Amer. Math. Soc. 340 (1993), no. 1, 253–272.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    R.R. Coifman, Distribution function inequalities for singular integrals, Proc. Nat. Acad. Sci. U.S.A. 69 (1972), 2838–2839.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    R.R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241–250.MathSciNetMATHGoogle Scholar
  7. [7]
    D. Cruz-Uribe, SFO, J.M. Martell and C. Pérez, Weights, Extrapolation and the Theory of Rubio de Francia, Operator Theory: Advances and Applications 215, Birkhäuser/Springer Basel AG, Basel, 2011.Google Scholar
  8. [8]
    J. García-Cuerva and J.L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland Math. Studies 116, North-Holland, Amsterdam, 1985.Google Scholar
  9. [9]
    S. Hruščev, A description of weights satisfying the A∞ condition of Muckenhoupt, Proc. Amer. Math. Soc. 90(2), 253–257, (1984).MathSciNetMATHGoogle Scholar
  10. [10]
    T. Hytönen and C. Pérez, Sharp weighted bounds involving A∞, Journal of Analysis and Partial Differential Equations, (2011), (to appear).Google Scholar
  11. [11]
    G.A., Karagulyan, Exponential estimates for the Calderón-Zygmund operator and related problems of Fourier series, Mat. Zametki 3 71, 398–41, (2002).Google Scholar
  12. [12]
    M. Lacey, An Ap–A∞ inequality for the Hilbert Transform, preprint. Available at http://arxiv.org/abs/1104.2199
  13. [13]
    A.K. Lerner,Weighted rearrangements inequalities for local sharp maximal functions, Trans. Amer. Math. Soc. (2004), 357, (6), 2445–2465.Google Scholar
  14. [14]
    A.K. Lerner, Sharp weighted norm inequalities for Littlewood-Paley operators and singular integrals, Advances in Mathematics (2011), 226, (5), 3912–3926.Google Scholar
  15. [15]
    A.K. Lerner, S. Ombrosi and C. Pérez, Sharp A1 bounds for Calderón-Zygmund operators and the relationship with a problem of Muckenhoupt and Wheeden, International Mathematics Research Notices, 2008, no. 6, Art. ID rnm161, 11 pp. 42B20.Google Scholar
  16. [16]
    A. Lerner, S. Ombrosi and C. Pérez, A1 bounds for Calderón-Zygmund operators related to a problem of Muckenhoupt and Wheeden, Mathematical Research Letters (2009), 16, 149–156.Google Scholar
  17. [17]
    A. Lerner, S. Ombrosi, C. Pérez, R. Torres and R. Trujillo-González, New maximal functions and multiple weights for the multilinear Calderón-Zygmund theory, Advances in Mathematics (2009), 220, 1222–1264.Google Scholar
  18. [18]
    C. Ortiz-Caraballo, Quadratic A1 bounds for commutators of singular integrals with BMO functions, to appear in Indiana Univ. Math. J. (2012).Google Scholar
  19. [19]
    C. Ortiz-Caraballo, Conmutadores de integrales singulares y pesos A1, Ph. D. Dissertation, (2011), Universidad de Sevilla.Google Scholar
  20. [20]
    C. Ortiz-Caraballo, C. Pérez and E. Rela, Local subexponential estimates for classical operators, Preprint 2011.Google Scholar
  21. [21]
    C. Pérez, A course on Singular Integrals and weights, to appear in Advanced Courses in Mathematics, CRM Barcelona, Birkhäuser editors.Google Scholar
  22. [22]
    C. Pérez, Weighted norm inequalities for singular integral operators, J. London Math. Soc., 49 (1994), 296–308.Google Scholar
  23. [23]
    C. Pérez, Endpoint Estimates for Commutators of Singular Integral Operators, Journal of Functional Analysis, (1) 128 (1995), 163–185.Google Scholar
  24. [24]
    C. Pérez, Sharp estimates for commutators of singular integrals via iterations of the Hardy-Littlewood maximal function, J. Fourier Anal. Appl. 3, 743–756, (1997).Google Scholar
  25. [25]
    C. Pérez and R. Trujillo-González, Sharp weighted estimates for vector-valued singular integrals operators and commutators, Tohoku Math. J. 55, 109–129, (2003).Google Scholar
  26. [26]
    J.M. Wilson, Weighted inequalities for the dyadic square function without dyadic A∞, Duke Math. J., 55(1), 19–50, (1987).MathSciNetMATHCrossRefGoogle Scholar
  27. [27]
    J.M. Wilson, Weighted inequalities for the continuous square function, Trans. Amer. Math. Soc., 314(2), 661–692, (1989).MathSciNetMATHCrossRefGoogle Scholar
  28. [28]
    J.M. Wilson, Weighted Littlewood-Paley theory and exponential-square integrability, volume 1924 of Lecture Notes in Mathematics. Springer, Berlin, (2008).Google Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  • Carmen Ortiz-Caraballo
    • 1
  • Carlos Pérez
    • 2
  • Ezequiel Rela
    • 2
  1. 1.Departamento de MatemáticasEscuela Politécnica Universidad de ExtremaduraCáceresSpain
  2. 2.Departamento De Análisis Matemático Facultad de MatemáticasUniversidad De SevillaSevillaSpain

Personalised recommendations