Banach Journal of Mathematical Analysis

Quantitative weighted bounds for the composition of Calderón–Zygmund operators

Guoen Hu

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Let T1, T2 be two Calderón–Zygmund operators, and let T1,b be the commutator of T1 with symbol bBMO(Rn). In this article, we establish the quantitative weighted bounds on Lp(Rn,w) with wAp(Rn) for the composite operator T1,bT2.

Article information

Banach J. Math. Anal., Volume 13, Number 1 (2019), 133-150.

Received: 17 March 2018
Accepted: 4 June 2018
First available in Project Euclid: 25 October 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)
Secondary: 47B33: Composition operators

weighted bound Calderón–Zygmund operator commutator bi-sublinear sparse operator sharp maximal operator


Hu, Guoen. Quantitative weighted bounds for the composition of Calderón–Zygmund operators. Banach J. Math. Anal. 13 (2019), no. 1, 133--150. doi:10.1215/17358787-2018-0019.

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  • [1] A. M. Alphonse, An end point estimate for maximal commutators, J. Fourier Anal. Appl. 6 (2000), no. 4, 449–456.
  • [2] C. Benea and F. Bernicot, Conservation de certaines propriétés à travers un contrôle épars d’un opérateur et applications au projecteur de Leray-Hopf, preprint, arXiv:1703.00228v1 [math.CA].
  • [3] S. M. Buckley, Estimates for operator norms on weighted spaces and reverse Jensen inequalities, Trans. Amer. Math. Soc. 340, no. 1 (1993), 253–272.
  • [4] N. Carozza and A. Passarelli di Napoli, Composition of maximal operators, Publ. Mat. 40 (1996), no. 2, 397–409.
  • [5] D. Chung, M. C. Pereyra, and C. Pérez, Sharp bounds for general commutators on weighted Lebesgue spaces, Trans. Amer. Math. Soc. 364, no. 3 (2012), 1163–1177.
  • [6] W. Damian, M. Hormozi, and K. Li, New bounds for bilinear Calderón-Zygmund operators and applications, Rev. Mat. Iberoam. 34 (2018), no. 3, 1177–1210.
  • [7] L. Grafakos, Modern Fourier Analysis, 2nd ed., Grad. Texts in Math. 250, Springer, New York, 2009.
  • [8] G. Hu and D. Li, A Cotlar type inequality for the multilinear singular integral operators and its applications, J. Math. Anal. Appl. 290 (2004), no. 2, 639–653.
  • [9] G. Hu and D. Yang, Weighted estimates for singular integral operators with nonsmooth kernels and applications, J. Aust. Math. Soc. 85 (2008), no. 3, 377–417.
  • [10] T. Hytönen, The sharp weighted bound for general Calderón-Zygmund operators, Ann. of Math. (2) 175 (2012), no. 3, 1473–1506.
  • [11] T. Hytönen and M. T. Lacey, The $A_{p}-A_{\infty}$ inequality for general Calderón-Zygmund operators, Indiana Univ. Math. J. 61 (2012), no. 6, 2041–2092.
  • [12] T. Hytönen and C. Pérez, Sharp weighted bounds involving $A_{\infty}$, Anal. PDE. 6 (2013), no. 4, 777–818.
  • [13] T. Hytönen and C. Pérez, The $L(\log L)^{\epsilon}$ endpoint estimate for maximal singular integral operators, J. Math. Anal. Appl. 428 (2015), no. 1, 605–626.
  • [14] T. Hytönen, C. Pérez, and E. Rela, Sharp reverse Hölder property for $A_{\infty}$ weights on spaces of homogeneous type, J. Funct. Anal. 263 (2012), no. 12, 3883–3899.
  • [15] S. G. Krantz and S. Li, Boundedness and compactness of integral operators on spaces of homogeneous type and applications, I, J. Math. Anal. Appl. 258 (2001), no. 2, 629–641.
  • [16] A. K. Lerner, On pointwise estimates involving sparse operators, New York J. Math. 22 (2016), 341–349.
  • [17] A. K. Lerner, A weak type estimate for rough singular integrals, preprint, arXiv:1705.07397v1 [math.CA].
  • [18] A. K. Lerner, S. Obmrosi, and I. P. Rivera-Ríos, On pointwise and weighted estimates for commutators of Calderón-Zygmund operators, Adv. Math. 319 (2017), 153–181.
  • [19] K. Li, Two weight inequalities for bilinear forms, Collect. Math. 68 (2017), no. 1, 129–144.
  • [20] K. Li, C. Pérez, I. P. Rivera-Ríos, and L. Roncal, Weighted norm inequalities for rough singular integral operators, preprint, arXiv:1701.05170v3 [math.CA].
  • [21] C. Pérez, Weighted norm inequalities for singular integral operators, J. Lond. Math. Soc. (2) 49 (1994), no. 2, 296–308.
  • [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), no. 5, 1355–1375.
  • [23] S. Petermichl, The sharp weighted bound for the Riesz transforms, Proc. Amer. Math. Soc. 136 (2008), no. 4, 1237–1249.
  • [24] M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces, Monogr. Textb. Pure Appl. Math. 146, Marcel Dekker, New York, 1991.
  • [25] M. J. Wilson, Weighted inequalities for the dyadic square function without dyadic $A_{\infty}$, Duke Math. J. 55 (1987), no. 1, 19–50.