Publicacions Matemàtiques

Weighted estimates for dyadic paraproducts and $\mathbf{t}$- multipliers with complexity $(m,n)$

Jean Carlo Moraes and María Christina Pereyra

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We extend the definitions of dyadic paraproduct and $t$-Haar multipliers to dyadic operators that depend on the complexity $(m,n)$, for $m$ and $n$ natural numbers. We use the ideas developed by Nazarov and Volberg to prove that the weighted $L^2(w)$\guio{norm} of a paraproduct with complexity~$(m,n)$, associated to a function $b\in \mathit{BMO}^d$, depends linearly on the $A^d_2$-characteristic of the weight~$w$, linearly on the $\mathit{BMO}^d$-norm of $b$, and polynomially on the complexity. This argument provides a new proof of the linear bound for the dyadic paraproduct due to Beznosova. We also prove that the $L^2$-norm of a $t$-Haar multiplier for any $t\in\mathbb{R}$ and weight~$w$ is a multiple of the square root of the $C^d_{2t}$-characteristic of $w$ times the square root of the $A^d_2$-characteristic of $w^{2t}$, and is polynomial in the complexity.

Article information

Publ. Mat. Volume 57, Number 2 (2013), 265-294.

First available in Project Euclid: 12 December 2013

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42C99: None of the above, but in this section
Secondary: 47B38: Operators on function spaces (general)

Operator-weighted inequalitiese dyadic paraproduct $A_p$-weights Haar multipliers


Moraes, Jean Carlo; Pereyra, María Christina. Weighted estimates for dyadic paraproducts and $\mathbf{t}$- multipliers with complexity $(m,n)$. Publ. Mat. 57 (2013), no. 2, 265--294.

Export citation


  • O. V. Beznosova, Bellman functions, paraproducts, Haar multipliers and weighted inequalities, Thesis (Ph.D.), The University of New Mexico (2008).
  • O. V. Beznosova, Linear bound for the dyadic paraproduct on weighted Lebesgue space $L_{2}(w)$, J. Funct. Anal. 255(4) (2008), 994\Ndash1007. \small\tt DOI: 10.1016/j.jfa.2008.04.025.
  • O. V. Beznosova, J. C. Moraes, and M. C. Pereyra, Sharp bounds for $t$-Haar multipliers in $L^2$, in: “Proceedings of the 9th International Conference on Harmonic Analysis and Partial Differential Equations (El Escorial, Spain, June 2012), Contemp. Math. (submitted).
  • O. Beznosova and A. Reznikov, Sharp estimates involving $A_\infty$ and $L\log L$ constants, and their applications to PDE, (Preprint 2011), \small\tt arXiv:1107.1885.
  • S. M. Buckley, Estimates for operator norms on weighted spaces and reverse Jensen inequalities, Trans. Amer. Math. Soc. 340(1) (1993), 253\Ndash272. \small\tt DOI: 10.2307/2154555.
  • D. Chung, Weighted inequalities for multivariable dyadicparaproducts, Publ. Mat. 55(2) (2011), 475\Ndash499. \small\tt DOI:\!\! 10.5565/ \small\tt PUBLMAT${}_{-}$55211${}_{-}$10.
  • D. Cruz-Uribe, J. M. Martell, and C. Pérez, Sharp weighted estimates for classical operators, Adv. Math. 229(1) (2012), 408\Ndash441. \small\tt DOI: 10.1016/j.aim.2011.08.013.
  • D. V. Cruz-Uribe, J. M. Martell, and C. Pérez, “Weights, extrapolation and the theory of Rubio de Francia”, Operator Theory: Advances and Applications 215,Birkhäuser/Springer Basel AG, Basel, 2011. \small\tt DOI: 10.1007/ \small\tt 978-3-0348-0072-3.
  • O. Dragičević, L. Grafakos, M. C. Pereyra, and S. Petermichl, Extrapolation and sharp norm estimates for classical operators on weighted Lebesgue spaces, Publ. Mat. 49(1) (2005), 73\Ndash91. \small\tt DOI: 10.5565/PUBLMAT${}_{-}$49105${}_{-}$03.
  • S. Hukovic, S. Treil, and A. Volberg, The Bellman functions and sharp weighted inequalities for square functions, in: “Complex analysis, operators, and related topics”, Oper. Theory Adv. Appl. 113, Birkhäuser, Basel, 2000, pp. 97\Ndash113.
  • T. Hytönen, The sharp weighted bound for general Calderón\guioZygmund operators, Ann. of Math. (2) 175(3) (2012), 1473\Ndash1506. \small\tt DOI: 10.4007/annals.2012.175.3.9.
  • T. P. Hytönen, M. T. Lacey, H. Martikainen, T. Orponen, M. C. Reguera, E. T. Sawyer, and I. Uriarte\guioTuero, Weak and strong type estimates for maximal truncations of Calderón-Zygmund operators on $A_{p}$ weighted spaces, J. Anal. Math. 118 (2012), 177\Ndash220. \small\tt DOI: 10.1007/s11854-012-0033-3.
  • T. Hytönen, C. Peréz, S. Treil, and A. Volberg, Sharp weighted estimates for dyadic shifts and the $A_{2}$ conjecture, J. Reine Angew. Math. (to appear).
  • N. H. Katz and M. C. Pereyra, Haar multipliers, paraproducts, and weighted inequalities, in: Analysis of divergence (Orono, ME, 1997), Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 1999, pp. 145\Ndash170.
  • M. Lacey, On the $A_2$ inequality for Calderón-Zygmund operators, in: “Recent Advances in Harmonic Analysis and Applications”, in honor of Konstantin Oskolkov, Springer Proceedings in Mathematics & Statistics 25, Springer, New York, 2013, pp. 235\Ndash242. \small\tt DOI: 10.1007/978-1-4614-4565-4${}_{-}$20.
  • M. Lacey, The linear bound in $A_{2}$ for Calderón-Zygmund operators: a survey, in: “Marcinkiewicz centenary volume”, Banach Center Publ. 95, Polish Acad. Sci. Inst. Math., Warsaw, 2011, pp. 97\Ndash114. \small\tt DOI: 10.4064/bc95-0-7.
  • M. T. Lacey, S. Petermichl, and M. C. Reguera, Sharp $A_{2}$ inequality for Haar shift operators, Math. Ann. 348(1) (2010), 127\Ndash141. \small\tt DOI: 10.1007/s00208-009-0473-y.
  • A. K. Lerner, Sharp weighted norm inequalities for Littlewood\guioPaley operators and singular integrals, Adv. Math. 226(5) (2011), 3912\Ndash3926. \small\tt DOI: 10.1016/j.aim.2010.11.009.
  • J. C. Moraes, Weighted estimates for dyadic operators with complexity, PhD Dissertation, University of New Mexico (2011).
  • J. C. Moraes, Weighted estimates for dyadic operators with complexity in geometrically doubling metric spaces, in preparation.
  • B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207\Ndash226. \small\tt DOI: 10.1090/S0002-9947-1972-0293384-6.
  • F. Nazarov, A. Reznikov, S. Treil, and A. Volberg, A Bellman function proof of the $L^2$ bump conjecture, J. Anal. Math. (to appear).
  • F. Nazarov, A. Reznikov, and A. Volberg, The proof of $A_2$ conjecture in a geometrically doubling metric space, Preprint (2011), available at \small\tt arXiv:\!\!\! 1106.1342v2.
  • F. Nazarov, S. Treil, and A. Volberg, The Bellman functions and two-weight inequalities for Haar multipliers, J. Amer. Math. Soc. 12(4) (1999), 909\Ndash928. \small\tt DOI: 10.1090/S0894-0347-99-00310-0.
  • F. Nazarov, S. Treil, and A. Volberg, Two weight inequalities for individual Haar multipliers and other well localized operators, Math. Res. Lett. 15(3) (2008), 583\Ndash597.
  • F. Nazarov and A. Volberg, Bellman function, polynomial estimates of weighted dyadic shifts, and $A_2$ conjecture, Preprint (2011).
  • F. Nazarov and A. Volberg, A simple sharp weighted estimate of the dyadic shifts on metric spaces with geometric doubling, Internat. Math. Res. Notices (2012). \small\tt DOI: 10.1093/imrn/rns159.
  • M. C. Pereyra, On the resolvents of dyadic paraproducts, Rev. Mat. Iberoamericana 10(3) (1994), 627\Ndash664.
  • M. C. Pereyra, Haar multipliers meet Bellman functions,Rev. Mat. Iberoam. 25(3) (2009), 799\Ndash840. \small\tt DOI: 10.4171/RMI/ \small\tt 584.
  • M. C. Pereyra, Sobolev spaces on Lipschitz curves, Pacific J. Math. 172(2) (1996), 553\Ndash589.
  • C. Pérez, S. Treil, and A. Volberg, On $A_2$ conjecture and corona decomposition of weights, Preprint (2010), available at \small\tt arXiv:\!\!\! 1006.2630.
  • S. Petermichl, Dyadic shifts and a logarithmic estimate for Hankel operators with matrix symbol, C. R. Acad. Sci. ParisSér. I Math. 330(6) (2000), 455\Ndash460. \small\tt DOI: 10.1016/S0764- \small\tt 4442(00)00162-2.
  • 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(5) (2007), 1355\Ndash1375. \small\tt DOI: 10.1353/ajm.2007.0036.
  • S. Petermichl, The sharp weighted bound for the Riesz transforms, Proc. Amer. Math. Soc. 136(4) (2008), 1237\Ndash1249. \small\tt DOI: 10.1090/S0002-9939-07-08934-4.
  • S. Petermichl and A. Volberg, Heating of the Ahlfors-Beurling operator: weakly quasiregular maps on the plane are quasiregular, Duke Math. J. 112(2) (2002), 281\Ndash305. \small\tt DOI: 10.1215/S0012-9074-02-11223-X.
  • S. Treil, Sharp $A_2$ estimates of Haar shifts via Bellman function, Preprint (2011), available at \small\tt arXiv:1105.2252.
  • A. Volberg, Bellman function technique in Harmonic Analysis, Lectures of INRIA Summer School in Antibes, June 2011, Preprint (2011), available at \small\tt arXiv:1106.3899.
  • J. Wittwer, A sharp estimate on the norm of the martingale transform, Math. Res. Lett. 7(1) (2000), 1\Ndash12.
  • J. Wittwer, A sharp estimate on the norm of the continuous square function, Proc. Amer. Math. Soc. 130(8) (2002),2335\Ndash2342 (electronic). \small\tt DOI: 10.1090/S0002-9939-02-06342-6.