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Weak and strong type estimates for maximal truncations of Calderón-Zygmund operators on A p weighted spaces

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Abstract

We show that for 1 < p < ∞, weight w ∈ A p , and any L 2-bounded Calderón-Zygmund operator T, there is a constant C T,p such that the weak- and strong-type inequalities

$${\left\| {{T_\natural}f} \right\|_{{L^{p,\infty }}(w)}} \le {C_{T,p}}{\left\| w \right\|_{{A_p}}}{\left\| f \right\|_{{L^p}(w )}}$$
$${\left\| {{T_\natural}f} \right\|_{{L^p}(w)}} \le {C_{T,p}}\left\| w \right\|_{{A_p}}^{\max \{ 1,{{(p - 1)}^{ - 1}}}{\left\| f \right\|_{{L^p}(w)}}$$

hold, where T ♮ denotes the maximal truncations of T and \({\left\| w \right\|_{{A_p}}}\) denotes the Muckenhoupt A p characteristic of w. These estimates are not improvable in the power of \({\left\| w \right\|_{{A_p}}}\). Our argument follows the outlines of those of Lacey-Petermichl-Reguera (Math. Ann. 2010) and Hytönen-Pérez-Treil-Volberg (arXiv, 2010) and contains new ingredients, including a weak-type estimate for certain duals of T ♮ and sufficient conditions for two-weight inequalities in L p for T ♮. Our proof does not rely upon extrapolation.

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References

  1. P. Auscher, S. Hofmann, C. Muscalu, T. Tao, and C. Thiele, Carleson measures, trees, extrapolation, and T (b) theorems, Publ. Mat. 46 (2002), 257–325.

    MathSciNet  MATH  Google Scholar 

  2. S. M. Buckley, Estimates for operator norms on weighted spaces and reverse Jensen inequalities, Trans. Amer. Math. Soc. 340 (1993), 253–272.

    Article  MathSciNet  MATH  Google Scholar 

  3. L. Carleson, On convergence and growth of partial sums of Fourier series, Acta Math. 116 (1966), 135–157.

    Article  MathSciNet  MATH  Google Scholar 

  4. D. Cruz-Uribe, J-M. Martell, and C. Pérez, Sharp weighted estimates for approximating dyadic operators, Electron. Res. Announc. Math. Sci. 17 (2010), 12–19.

    MathSciNet  MATH  Google Scholar 

  5. S. V. Hruščev, A description of weights satisfying the A∞ condition of Muckenhoupt, Proc. Amer. Math. Soc. 90 (1984), 253–257.

    MathSciNet  MATH  Google Scholar 

  6. R. Hunt, B. Muckenhoupt, and R. Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc. 176 (1973), 227–251.

    Article  MathSciNet  MATH  Google Scholar 

  7. T. Hytönen, The sharp weighted bound for general Calderon-Zygmund operators, Ann. of Math. (2) 175 (2012), 1473–1506.

    Article  MathSciNet  MATH  Google Scholar 

  8. T. Hytönen, M. T. Lacey, M-C. Reguera, and A. Vagharshakyan, Weak and Strong-type estimates for Haar shift operators: sharp power on the A p characteristic, arXiv:0911.0713v2.

  9. T. Hytönen, M. T. Lacey, M-C Reguera, E. T. Sawyer, I. Uriarte-Tuero, Ignacio, and A. Vagharshakyan, Weak and strong type A p estimates for Caldern-Zygmund operators, arXiv:1006.2530v3.

  10. T. Hytönen and C. Pérez, Sharp weighted bounds involving A ∞, Anal. PDE, to appear.

  11. T. Hytönen, C. Pérez, S. Treil, and A. Volberg, Sharp weighted estimates for dyadic shifts and A 2 conjecture, J. Reine Angew. Math., to appear.

  12. M. T. Lacey, S. Petermichl, and M-C. Reguera, Sharp A 2 inequality for Haar shift operators, Math. Ann. 348 (2010), 127–141.

    Article  MathSciNet  MATH  Google Scholar 

  13. M. T. Lacey, E. T. Sawyer, and I. Uriarte-Tuero, Two weight inequalities for discrete positive operators, arXiv:0911.3437v4.

  14. M. T. Lacey, E. T. Sawyer, and I. Uriarte-Tuero, A characterization of two weight norm inequalities for maximal singular integrals with one doubling measure, arXiv:0807.0246v8.

  15. M. T. Lacey, E. T. Sawyer, and I. Uriarte-Tuero, Two weight inequalities for maximal truncations of dyadic Calderón-Zygmund operators, arXiv:0911.3920v3.

  16. M. T. Lacey and C. Thiele, A proof of boundedness of the Carleson operator, Math. Res. Lett. 7 (2000), 361–370.

    MathSciNet  MATH  Google Scholar 

  17. A. K. Lerner, Sharp weighted norm inequalities for Littlewood-Paley operators and singular integrals, Adv. Math. 226 (2011), 3912–3926.

    Article  MathSciNet  MATH  Google Scholar 

  18. A. K. Lerner, On some weighted norm inequalities for Littlewood-Paley operators, Illinois J. Math. 52 (2007), 653–666.

    MathSciNet  Google Scholar 

  19. A. K. Lerner and S. Ombrosi, An extrapolation theorem with applications to weighted estimates for singular integrals, J. Funct. Anal. 262 (2012), 4475–4487.

    Article  MathSciNet  MATH  Google Scholar 

  20. F. Nazarov, S. Treil, and A. Volberg, Accretive system Tb-theorems on nonhomogeneous spaces, Duke Math. J. 113 (2002), 259–312.

    Article  MathSciNet  MATH  Google Scholar 

  21. F. Nazarov, S. Treil, and A. Volberg, The Bellman functions and two-weight inequalities for Haar multipliers, J. Amer. Math. Soc. 12 (1999), 909–928.

    Article  MathSciNet  MATH  Google Scholar 

  22. S. Petermichl, The sharp bound for the Hilbert transform on weighted Lebesgue spaces in terms of the classical A p characteristic, Amer. J. Math. 129 (2007), 1355–1375.

    Article  MathSciNet  MATH  Google Scholar 

  23. E. T. Sawyer, A characterization of a two-weight norm inequality for maximal operators, Studia Math. 75 (1982), 1–11.

    MathSciNet  MATH  Google Scholar 

  24. E. Sawyer, A two weight weak-type inequality for fractional integrals, Trans. Amer. Math. Soc. 281 (1984), 339–345.

    Article  MathSciNet  MATH  Google Scholar 

  25. E. T. Sawyer, A characterization of two weight norm inequalities for fractional and Poisson integrals, Trans. Amer. Math. Soc. 308, (1988), 533–545.

    Article  MathSciNet  MATH  Google Scholar 

  26. M. J. Wilson, Weighted inequalities for the dyadic square function without dyadic A ∞, Duke Math. J. 55 (1987), 19–50.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Helsinki, FI-00014, Helsinki, Finland

    Tuomas P. Hytönen, Henri Martikainen & Tuomas Orponen

  2. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Michael T. Lacey & Maria Carmen Reguera

  3. Department of Mathematics & Statistics, Mcmaster University, Hamilton, Ontario, Canada, L8S 4K1

    Eric T. Sawyer

  4. Department of Mathematics, Michigan State University, East Lansing, MI, 48824, USA

    Ignacio Uriarte-Tuero

Authors
  1. Tuomas P. Hytönen
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  2. Michael T. Lacey
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  3. Henri Martikainen
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  4. Tuomas Orponen
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  5. Maria Carmen Reguera
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  6. Eric T. Sawyer
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  7. Ignacio Uriarte-Tuero
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Corresponding author

Correspondence to Tuomas P. Hytönen.

Additional information

Research of TPH and HM supported by the Academy of Finland grants 130166, 133264 and 218418

Research of MTL, and MCR supported in part by NSF grant 0968499.

Research of TO supported by the Finnish Centre of Excellence in Analysis and Dynamics Research.

Research of ETS supported in part by NSERC

Research of IU-T supported in part by the NSF, through grant DMS-0901524.

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Hytönen, T.P., Lacey, M.T., Martikainen, H. et al. Weak and strong type estimates for maximal truncations of Calderón-Zygmund operators on A p weighted spaces. JAMA 118, 177–220 (2012). https://doi.org/10.1007/s11854-012-0033-3

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  • DOI: https://doi.org/10.1007/s11854-012-0033-3

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