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
Assaad, J.: Riesz transforms associated to Schrödinger operators with negative potentials. Publ. Mat. 55(1), 123–150 (2011)
Assaad, J., Ouhabaz, E.M.: Riesz transforms of Schrödinger operators on manifolds. J. Geom. Anal. 22, 1108–1136 (2012)
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)
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)
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)
Bernicot, F., Frey, D., Petermichl, S.: Sharp weighted norm estimates beyond Calderón–Zygmund theory. Anal. PDE 9(5), 1079–1113 (2016)
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)
Buckley, S.M.: Estimates for operator norms on weighted spaces and reverse Jensen inequalities. Trans. Am. Math. Soc. 340, 253–272 (1993)
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)
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)
Christ, M.: A $Tb$ theorem with remarks on analytic capacity and the Cauchy integral. Colloq. Math. 61, 601–628 (1990)
Coulhon, T., Duong, X.T.: Riesz transforms for $1\le p \le 2$. Trans. Am. Math. Soc. 351(3), 1151–1169 (1999)
Duong, X.T., McIntosh, A.: Singular integral operators with non-smooth kernels on irregular domains. Rev. Mat. Iberoam. 15, 233–265 (1999)
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)
Hebisch, W., Saloff-Coste, L.: On the relation between elliptic and parabolic Harnack inequalities. Ann. Inst. Fourier (Grenoble) 51, 1437–1481 (2001)
Hytönen, T.: The sharp weighted bound for general Calderón–Zygmund operators. Ann. Math. (2) 175(3), 1473–1506 (2012)
Hytönen, T., Kairema, A.: Systems of dyadic cubes in a doubling metric space. Colloq. Math. 126(1), 1–33 (2012)
Killip, R., Visan, M., Zhang, X.: Riesz transforms outside a convex obstacle. Int. Math. Res. Not. IMRN 2016, 5875–5921 (2016)
Lacey, M.: An elementary proof of the $A_2$ bound. Isr. J. Math. 217(1), 181–195 (2017)
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)
Lerner, A.K.: A simple proof of the $A_2$ conjecture. Int. Math. Res. Not. 14, 3159–3170 (2013)
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)
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Xuan Thinh Duong was supported by Australian Research Council through the ARC grant DP160100153.
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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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DOI: https://doi.org/10.1007/s00209-019-02416-4