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Weighted Norm Inequalities for Rough Singular Integral Operators

  • Kangwei LiEmail author
  • Carlos Pérez
  • Israel P. Rivera-Ríos
  • Luz Roncal
Article

Abstract

In this paper we provide weighted estimates for rough operators, including rough homogeneous singular integrals \(T_\Omega \) with \(\Omega \in L^\infty (\mathbb {S}^{n-1})\) and the Bochner–Riesz multiplier at the critical index \(B_{(n-1)/2}\). More precisely, we prove qualitative and quantitative versions of Coifman–Fefferman type inequalities and their vector-valued extensions, weighted \(A_p-A_\infty \) strong and weak type inequalities for \(1<p<\infty \), and \(A_1-A_\infty \) type weak (1, 1) estimates. Moreover, Fefferman–Stein type inequalities are obtained, proving in this way a conjecture raised by the second-named author in the 1990s. As a corollary, we obtain the weighted \(A_1-A_\infty \) type estimates. Finally, we study rough homogenous singular integrals with a kernel involving a function \(\Omega \in L^q(\mathbb {S}^{n-1})\), \(1<q<\infty \), and provide Fefferman–Stein inequalities too. The arguments used for our proofs combine several tools: a recent sparse domination result by Conde–Alonso et al. (Anal PDE 10(5):1255–1284, 2017), results by the first author (Collect Math 68:129–144, 2017), suitable adaptations of Rubio de Francia algorithm, the extrapolation theorems for \(A_{\infty }\) weights (Cruz-Uribe et al. in J Funct Anal 213:412–439, 2004, Curbera et al. in Adv Math 203:256–318, 2006), and ideas contained in previous works by Seeger (J Am Math Soc 9:95–105 1996) and Fan and Sato (Tohoku Math J 53:265–284, 2001).

Keywords

Rough operators Weights Fefferman–Stein inequalities Sparse operators Rubio de Francia algorithm 

Mathematics Subject Classification

Primary 42B20 Secondary 42B25 42B15 

Notes

Acknowledgements

We are very grateful to the referee for pointing out that the quantitative weak type estimate in Theorem 1.11 is a direct consequence of the sparse bound and for providing a proof of Theorem C.1. We thank Francesco Di Plinio for sending us a preprint of [9] and for some useful comments calling our attention to some results from the paper by David Beltran [3]. We would also like to thank David Beltran for telling us his qualitative proof of Theorem 1.12.

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Copyright information

© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  • Kangwei Li
    • 1
    Email author
  • Carlos Pérez
    • 2
  • Israel P. Rivera-Ríos
    • 3
  • Luz Roncal
    • 1
  1. 1.BCAM, Basque Center for Applied MathematicsBilbaoSpain
  2. 2.Departamento de MatemáticasUniversidad del País Vasco UPV/EHU, IKERBASQUE, Basque Foundation for Science, and BCAM, Basque Center for Applied MathematicsBilbaoSpain
  3. 3.Departamento de MatemáticasUniversidad del País Vasco UPV/EHU and BCAMBilbaoSpain

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