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

  • The Anh Bui
  • Xuan Thinh DuongEmail author
Article
  • 16 Downloads

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.

Keywords

Heat kernels Singular operators Weighted estimates 

Mathematics Subject Classification

58J35 42B20 

Notes

Acknowledgements

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

References

  1. 1.
    Assaad, J.: Riesz transforms associated to Schrödinger operators with negative potentials. Publ. Mat. 55(1), 123–150 (2011)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Assaad, J., Ouhabaz, E.M.: Riesz transforms of Schrödinger operators on manifolds. J. Geom. Anal. 22, 1108–1136 (2012)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bernicot, F., Frey, D., Petermichl, S.: Sharp weighted norm estimates beyond Calderón–Zygmund theory. Anal. PDE 9(5), 1079–1113 (2016)MathSciNetCrossRefGoogle 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)CrossRefGoogle 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)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle 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)CrossRefGoogle Scholar
  11. 11.
    Christ, M.: A $Tb$ theorem with remarks on analytic capacity and the Cauchy integral. Colloq. Math. 61, 601–628 (1990)CrossRefGoogle 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)CrossRefGoogle 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)MathSciNetCrossRefGoogle 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)zbMATHGoogle 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)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Hytönen, T.: The sharp weighted bound for general Calderón–Zygmund operators. Ann. Math. (2) 175(3), 1473–1506 (2012)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Hytönen, T., Kairema, A.: Systems of dyadic cubes in a doubling metric space. Colloq. Math. 126(1), 1–33 (2012)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Killip, R., Visan, M., Zhang, X.: Riesz transforms outside a convex obstacle. Int. Math. Res. Not. IMRN 2016, 5875–5921 (2016)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Lacey, M.: An elementary proof of the $A_2$ bound. Isr. J. Math. 217(1), 181–195 (2017)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Lerner, A.K.: A simple proof of the $A_2$ conjecture. Int. Math. Res. Not. 14, 3159–3170 (2013)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsMacquarie UniversityMacquarieAustralia

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