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)


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].


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


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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

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