Sharp weighted norm inequalities for singular integrals with non–smooth kernels

Abstract

In this paper, we prove the sharp weighted bound for certain singular integrals which have non-smooth kernels and do not belong to the class of standard Calderón–Zygmund operators. Our assumptions are weaker than those known in literature, since in particular we do not assume the Cotlar type inequality condition. Applications include sharp weighted estimates for the Riesz transforms associated to the Dirichlet Laplacians on open connected domains, the Riesz transforms associated to the Schrödinger operators with real potentials on the Euclidean spaces, the Riesz transforms associated to the degenerate Schrödinger operators and the Riesz transforms associated to the Schrödinger operators with inverse square potentials.

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References

  1. 1.

    Assaad, J.: Riesz transforms associated to Schrödinger operators with negative potentials. Publ. Mat. 55(1), 123–150 (2011)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Assaad, J., Ouhabaz, E.M.: Riesz transforms of Schrödinger operators on manifolds. J. Geom. Anal. 22, 1108–1136 (2012)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Auscher, P., Ben Ali, B.: Maximal inequalities and Riesz transform estimates on $L^p$ spaces for Schrödinger operators with nonnegative potentials. Ann. Inst. Fourier (Grenoble) 57, 1975–2013 (2007)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Auscher, P., Martell, J.M.: Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part I: general operator theory and weights. Adv. Math. 212, 225–276 (2007)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Auscher, P., Coulhon, T., Duong, X.T., Hofmann, S.: Riesz transform on manifolds and heat kernel regularity. Ann. Sci. École Norm. Sup. 37, 911–957 (2004)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Bernicot, F., Frey, D., Petermichl, S.: Sharp weighted norm estimates beyond Calderón–Zygmund theory. Anal. PDE 9(5), 1079–1113 (2016)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Blunck, S., Kunstmann, P.C.: Calderón–Zygmund theory for non-integral operators and the $H^{\infty }$ functional calculus. Rev. Mat. Iberoam. 19, 919–942 (2003)

    MATH  Google Scholar 

  8. 8.

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

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Bui, T.A., Conde-Alonso, J.M., Duong, X.T., Hormozi, M.: A note on weighted bounds for singular operators with nonsmooth kernels. Studia Math. 236(3), 245–269 (2017)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Bui, T.A., D’Ancona, P., Duong, X.T., Li, J., Ly, F.K.: Weighted estimates for powers and smoothing estimates of Schrödinger operators with inverse-square potentials. J. Differ. Equ. 262(3), 2771–2807 (2016)

    MATH  Google Scholar 

  11. 11.

    Christ, M.: A $Tb$ theorem with remarks on analytic capacity and the Cauchy integral. Colloq. Math. 61, 601–628 (1990)

    MATH  Google Scholar 

  12. 12.

    Coulhon, T., Duong, X.T.: Riesz transforms for $1\le p \le 2$. Trans. Am. Math. Soc. 351(3), 1151–1169 (1999)

    MATH  Google Scholar 

  13. 13.

    Duong, X.T., McIntosh, A.: Singular integral operators with non-smooth kernels on irregular domains. Rev. Mat. Iberoam. 15, 233–265 (1999)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Duong, X.T., McIntosh, A.: The $L^p$ boundedness of Riesz transforms associated with divergence form operators. Joint Australian-Taiwanese Workshop on Analysis and Applications. Proc. Centre Math. Appl. 37, 15–25 (1999)

    MATH  Google Scholar 

  15. 15.

    Hebisch, W., Saloff-Coste, L.: On the relation between elliptic and parabolic Harnack inequalities. Ann. Inst. Fourier (Grenoble) 51, 1437–1481 (2001)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Hytönen, T.: The sharp weighted bound for general Calderón–Zygmund operators. Ann. Math. (2) 175(3), 1473–1506 (2012)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Hytönen, T., Kairema, A.: Systems of dyadic cubes in a doubling metric space. Colloq. Math. 126(1), 1–33 (2012)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Killip, R., Visan, M., Zhang, X.: Riesz transforms outside a convex obstacle. Int. Math. Res. Not. IMRN 2016, 5875–5921 (2016)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Lacey, M.: An elementary proof of the $A_2$ bound. Isr. J. Math. 217(1), 181–195 (2017)

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Lerner, A.K.: A pointwise estimate for the local sharp maximal function with applications to singular integrals. Bull. Lond. Math. Soc. 42(5), 843–856 (2010)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Lerner, A.K.: A simple proof of the $A_2$ conjecture. Int. Math. Res. Not. 14, 3159–3170 (2013)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Liskevich, V., Sobol, Z.: Estimates of integral kernels for semigroups associated with second order elliptic operators with singular coefficients. Potential Anal. 18, 359–390 (2003)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

Xuan Thinh Duong was supported by Australian Research Council through the ARC grant DP160100153.

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Correspondence to Xuan Thinh Duong.

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Bui, T.A., Duong, X.T. Sharp weighted norm inequalities for singular integrals with non–smooth kernels. Math. Z. 295, 1733–1750 (2020). https://doi.org/10.1007/s00209-019-02416-4

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Keywords

  • Heat kernels
  • Singular operators
  • Weighted estimates

Mathematics Subject Classification

  • 58J35
  • 42B20