Journal of the Mathematical Society of Japan

Generalized commutators of multilinear Calderón–Zygmund type operators

Qingying XUE and Jingquan YAN

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Let $T$ be an $m$-linear Calderón–Zygmund operator with kernel $K$ and $T^*$ be the maximal operator of $T$. Let $S$ be a finite subset of $Z^+\times \{1,\dots,m\}$ and denote $d\vec{y}=dy_1\cdots dy_m$. Define the commutator $T_{\vec{b},S}$ of $T$, and $T^*_{\vec{b},S}$ of $T^*$ by $T_{\vec{b},S}(\vec{f})(x)= \int_{\mathbb{R}^{nm}}\prod_{(i,j)\in S}(b_i(x)-b_i(y_j)) K(x,y_{1},\dots,y_{m}) \prod_{j=1}^mf_j(y_j)d\vec{y}$ and $T^*_{\vec{b},S}(\vec{f})(x)= \sup_{\delta>0}\big|\int_{{\sum_{j=1}^m|x-y_j|^2>\delta^2}} \prod_{(i,j)\in S}(b_i(x)-b_i(y_j))K(x,y_{1},\dots,y_{m}) \prod_{j=1}^mf_{j}(y_j)d\vec{y}\big|$. These commutators are reflexible enough to generalize several kinds of commutators which already existed. We obtain the weighted strong and endpoint estimates for $T_{\vec{b},S}$ and $T^*_{\vec{b},S}$ with multiple weights. These results are based on an estimate of the Fefferman–Stein sharp maximal function of the commutators, which is proved in a pretty much more organized way than some known proofs. Similar results for the commutators of vector-valued multilinear Calderón–Zygmund operators are also given.

Article information

J. Math. Soc. Japan Volume 68, Number 3 (2016), 1161-1188.

First available in Project Euclid: 19 July 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)
Secondary: 42B25: Maximal functions, Littlewood-Paley theory

multilinear Calderón–Zygmund operators Commutators Multiple weights


XUE, Qingying; YAN, Jingquan. Generalized commutators of multilinear Calderón–Zygmund type operators. J. Math. Soc. Japan 68 (2016), no. 3, 1161--1188. doi:10.2969/jmsj/06831161.

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  • J. Alvarez, R. J. Bagby, D. S. Kurtz and C. Pérez, Weighted estimates for commutators of linear operators, Studia Math., 104 (1993), 195–209.
  • J. Alvarez and C. Pérez, Estimates with $A_{\infty}$ weights for various singular integral operators, Boll. Un. Mat. Ital. A, 8 (7) (1994), 123–133.
  • R. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functionsand singular integrals, Studia Math., 51 (1974), 241–250.
  • R. R. Coifman and Y. Meyer, On commutators of singular integrals and bilinear singular integrals, Trans. Amer. Math. Soc., 212 (1975), 315–331.
  • R. R. Coifman and Y. Meyer, Commutateurs d'intégrales singulières et opérateurs multilinéaires, Ann. Inst. Fourier, Grenoble., 28 (1978), 177–202.
  • R. R. Coifman and Y. Meyer, Au-delà des opérateurs pseudo-différentiels, Asterisque, 57, Société Mathématique de France, Paris, 1978.
  • R. Coifman, R. Rochberg and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math., 103 (1976), 611–635.
  • D. Cruz-Uribe, J. M. Martell and C. Pérez, Extrapolation from $A_{\infty}$ weights and applications, J. Funct. Anal., 213 (2004), 412–439.
  • L. Grafakos, L. Liu, C. Pérez and R. H. Torres, The multilinear strong maximal function, J. Geom. Anal., 21 (2011), 118–149.
  • L. Grafakos and J. Martell, Extrapolation of Weighted Norm Inequalities for Multivariable Operators and Applications, J. Geom. Anal., 14 (2004), 19–46.
  • L. Grafakos and R. H. Torres, Multilinear Calderón–Zygmund theory, Adv. Math., 165 (2002), 124–164.
  • T. Hytonen and C. Pérez, Sharp weighted bounds involving $A_{\infty}$, Analysis and P.D.E., 6 (2013), 777–818.
  • S. Janson, Mean oscillation and commutators of singular integral operators, Ark. Mat., 16 (1978), 263–270.
  • C. E. Kenig and E. M. Stein, Multilinear estimates and fractional integration, Math. Res. Lett., 6 (1999), 1–15.
  • A. K. Lerner, S. Ombrosi, C. Pérez, R. H. Torres and R. Trujillo-González, New maximal functions and multiple weights for the multilinear Calderón–Zygmund theory, Adv. Math., 220 (2009), 1222–1264.
  • J. Mateu, J. Orobitg, C. Pérez and J. Verdera, New estimates for the maximal singular integral, Int. Math. Res. Not. IMRN, (2010) Vol.,2010, 3658–3722.
  • A. Micheal Alphonse, An end point estimate for maximal commutators, J. Fourier Anal. Appl., 6 (2000), 449–456.
  • B. Moukenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Am. Math. Soc., 165 (1972), 207–226.
  • C. Ortiz-Caraballo, Quadratic $A^1$ bounds for commutators of singular integrals with BMO functions, Indiana Univ. Math. J., 60 (2011), 2107–2130.
  • C. Ortiz-Caraballo, C. Pérez and E. Rela, Improving bounds for singular operator via Sharp Reverse Hólder Inequality for $A_{\infty}$, Advances in Harmonic Analysis and Operator Theory. Birkhäuser OT series, 229 (2013), 303–321.
  • C. Pérez, End point estimates for commutators of singular integral operators, J. Funct. Anal., 128 (1995), 163–185.
  • C. Pérez, Sharp estimates for commutators of singular integrals via iterations of the Hardy-Littlewood maximal function, J. Fourier Anal. Appl., 3 (1997), 743–756.
  • C. Pérez, A course on Singular Integrals and weights, Adv. Courses Math., CRM Barcelona, Birkhäuser editors.
  • C. Pérez and G. Pradolini, Sharp weighted endpoint estimates for commutators of singular integral operators, Michigan Math. J., 49 (2001), 23–37.
  • C. Pérez, G. Pradolini, R. H. Torres and R. Trujillo-González, End-point estimates for iterated commutators of multilinears ingular integrals, Bull. Lond. Math. Soc., 46 (2014), 26–42.
  • C. Pérez and R. H. Torres, Sharp maximal function estimates for multilinear singular integrals, Harmonic Analysis at Mount Holyoke, Contemp. Math., 320 (2003), 323–331.
  • C. Pérez and R. Trujillo-González, Sharp weighted estimates for multilinear commutators, J. London Math. Soc., 65 (2002), 672–692.
  • C. Pérez and R. Trujillo-González, Sharp weighted estimates for vector-valued singular integral operators and commutators, Tohoku Math. J. (2), 55 (2003), 109–129.
  • M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces, Monographs and Textbooks in Pure and Applied Math., 146, Marcel Dekker, New York, 1991.
  • J. L. Rubio de Francia, Factorization theory and $A_p$ weights, Amer. J. Math., 106 (1984), 533–547.
  • J. L. Rubio de Francia, F. J. Ruiz and J. L. Torrea, Calderón–Zygmund theory for operator-valued kernels, Adv. Math., 62 (1986), 7–48.
  • C. Segovia and J. L. Torrea, Weighted inequalities for commutators of fractional and singular integrals, Publ. Mat., 35 (1991), 209–235.
  • Z. Si and Q. Xue, weighted estimats for commutators of vector-valued maximal multilinear operators, Nonlinear Analysis Series A: TMA, 96 (2014), 96–108.
  • L. Tang, Weighted estimates for vector-valued commutators of multilinear operators, Proc. Roy. Soc. Edinburgh Section A, 138 (2008), 897–922.
  • J. M. Wilson, Weighted inequalities for the dyadic square function without dyadic $A_{\infty}$, Duke Math. J., 55 (1987), 19–50.
  • Q. Xue, Weighted estimates for the iterated commutators of multilinear maximal and fractonal type operators, Studia. Math., 217 (2013), 97–122
  • P. Zhang, Weighted estimates for maximal multilinear commutators, Math. Nachr., 279 (2006), 445–462.