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Two-weight commutator estimates: general multi-parameter framework. (English) Zbl 1471.42029

Let \(T_i\) be an \(m_i\)-parameter Calderón-Zygmund operator, \(1\le i\le k\), and let \(b\) belong to a suitable weighted little product BMO space. The author considered the following two weight commutator estimates \[ [T_1, [T_2, \dots, [b, T_k]]]: L^p(\mu)\mapsto L^p(\lambda), \] where \(1<p<\infty\), \(\mu, \lambda\in A_p\) and \(\nu=\mu^{\frac 1p}\lambda^{-\frac 1p}\) is the Bloom weight. In other words, the author wanted to provide the multi-parameter analog of the recent work by K. Li et al. [J. Geom. Anal. 30, No. 3, 3181–3203 (2020; Zbl 1440.42063)].
The main result of the paper under review is that, if for all \(1\le i\le k\), either \(T_i\) is paraproduct free or \(m_i\le 2\), then \(\|[T_1, [T_2, \dots, [b, T_k]]]\|_{L^p(\mu)\to L^p(\lambda)}\) is controlled by suitable weighted little product BMO norm of \(b\).
In the recent work [E. Airta et al., “Some new weighted estimates on product spaces”, Preprint, arXiv:1910.12546], the restrictions on \(T_i\), i.e., either \(T_i\) is paraproduct free or \(m_i\le 2\), are removed.

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B35 Function spaces arising in harmonic analysis

Citations:

Zbl 1440.42063
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References:

[1] A. Barron, J. M. Conde-Alonso, Y. Ou, and G. Rey, Sparse domination and the strong maximal function,Adv. Math.345(2019), 1-26.DOI: 10.1016/ j.aim.2019.01.007. · Zbl 1410.42020
[2] A. Barron and J. Pipher, Sparse domination for bi-parameter operators using square functions, Preprint (2017).arXiv:1709.05009.
[3] S. Bloom, A commutator theorem and weighted BMO,Trans. Amer. Math. Soc.292(1)(1985), 103-122.DOI: 10.2307/2000172. · Zbl 0578.42012
[4] S.-Y. A. Chang and R. Fefferman, A continuous version of duality ofH1with BMO on the bidisc,Ann. of Math. (2)112(1)(1980), 179-201.DOI: 10.2307/ 1971324. · Zbl 0451.42014
[5] S.-Y. A. Chang and R. Fefferman, Some recent developments in Fourier analysis andHp-theory on product domains,Bull. Amer. Math. Soc. (N.S.)12(1) (1985), 1-43.DOI: 10.1090/S0273-0979-1985-15291-7. · Zbl 0557.42007
[6] R. R. Coifman, R. Rochberg, and G. Weiss, Factorization theorems for Hardy spaces in several variables,Ann. of Math. (2)103(3)(1976), 611-635.DOI: 10. 2307/1970954. · Zbl 0326.32011
[7] D. Cruz-Uribe, J. M. Martell, and C. P´erez, Extrapolation fromA∞weights and applications,J. Funct. Anal.213(2)(2004), 412-439.DOI: 10.1016/j.jfa. 2003.09.002. · Zbl 1052.42016
[8] L. Dalenc and Y. Ou, Upper bound for multi-parameter iterated commutators, Publ. Mat.60(1)(2016), 191-220.DOI: 10.5565/PUBLMAT−60116−07. · Zbl 1333.42018
[9] G. David and J.-L. Journ´e, A boundedness criterion for generalized Calder´on- Zygmund operators,Ann. of Math. (2)120(2)(1984), 371-397.DOI: 10.2307/ 2006946. · Zbl 0567.47025
[10] X. T. Duong, J. Li, Y. Ou, J. Pipher, and B. D. Wick, Commutators of multiparameter flag singular integrals and applications,Anal. PDE12(5)(2019), 1325-1355.DOI: 10.2140/apde.2019.12.1325. · Zbl 1405.42041
[11] C. Fefferman and E. M. Stein, Some maximal inequalities,Amer. J. Math. 93(1)(1971), 107-115.DOI: 10.2307/2373450. · Zbl 0222.26019
[12] R. Fefferman and E. M. Stein, Singular integrals on product spaces,Adv. in Math.45(2)(1982), 117-143.DOI: 10.1016/S0001-8708(82)80001-7. · Zbl 0517.42024
[13] S. H. Ferguson and M. T. Lacey, A characterization of productBMOby commutators,Acta Math.189(2)(2002), 143-160.DOI: 10.1007/BF02392840. · Zbl 1039.47022
[14] S. H. Ferguson and C. Sadosky, Characterizations of bounded mean oscillation on the polydisk in terms of Hankel operators and Carleson measures,J. Anal. Math.81(2000), 239-267.DOI: 10.1007/BF02788991. · Zbl 0988.32002
[15] A. Grau de la Herr´an, Comparison ofT1 conditions for multi-parameter operators,Proc. Amer. Math. Soc.144(6)(2016), 2437-2443.DOI: 10.1090/proc/ 12891. · Zbl 1350.42024
[16] I. Holmes, M. T. Lacey, and B. D. Wick, Bloom’s inequality: commutators in a two-weight setting,Arch. Math. (Basel)106(1)(2016), 53-63.DOI: 10.1007/ s00013-015-0840-8. · Zbl 1342.42005
[17] I. Holmes, M. T. Lacey, and B. D. Wick, Commutators in the two-weight setting,Math. Ann.367(1-2)(2017), 51-80.DOI: 10.1007/s00208-016-1378-1. · Zbl 1364.42017
[18] I. Holmes, S. Petermichl, and B. D. Wick, Weighted little bmo and two-weight inequalities for Journ´e commutators,Anal. PDE11(7)(2018), 1693-1740.DOI: 10.2140/apde.2018.11.1693. · Zbl 1395.42064
[19] I. Holmes and B. D. Wick, Two weight inequalities for iterated commutators with Calder´on-Zygmund operators,J. Operator Theory79(1)(2018), 33-54. DOI: 10.7900/jot. · Zbl 1424.42024
[20] T. P. Hyt¨onen, The sharp weighted bound for general Calder´on-Zygmund operators,Ann. of Math. (2)175(3)(2012), 1473-1506.DOI: 10.4007/annals. 2012.175.3.9. · Zbl 1250.42036
[21] T. P. Hyt¨onen, The Holmes-Wick theorem on two-weight bounds for higher order commutators revisited,Arch. Math. (Basel)107(4)(2016), 389-395. DOI: 10.1007/s00013-016-0956-5. · Zbl 1354.42021
[22] T. P. Hyt¨onen, The two-weight inequality for the Hilbert transform with general measures,Proc. Lond. Math. Soc. (3)117(3)(2018), 483-526.DOI: 10. 1112/plms.12136. · Zbl 1420.42010
[23] T. P. Hyt¨onen, TheLp-to-Lqboundedness of commutators with applications to the Jacobian operator, Preprint (2018).arXiv:1804.11167.
[24] J.-L. Journ´e, Calder´on-Zygmund operators on product spaces,Rev. Mat. Iberoamericana1(3)(1985), 55-91.DOI: 10.4171/RMI/15. · Zbl 0634.42015
[25] I. Kunwar and Y. Ou, Two-weight inequalities for multilinear commutators, New York J. Math.24(2018), 980-1003. · Zbl 1403.42011
[26] M. T. Lacey, Two-weight inequality for the Hilbert transform: a real variable characterization, II,Duke Math. J.163(15)(2014), 2821-2840.DOI: 10.1215/ 00127094-2826799. · Zbl 1312.42010
[27] M. T. Lacey, S. Petermichl, J. C. Pipher, and B. D. Wick, Multiparameter Riesz commutators,Amer. J. Math.131(3)(2009), 731-769.DOI: 10.1353/ajm. 0.0059. · Zbl 1170.42003
[28] M. T. Lacey, S. Petermichl, J. C. Pipher, and B. D. Wick, Iterated Riesz commutators: a simple proof of boundedness, in:“Harmonic Analysis and Partial Differential Equations”, Contemp. Math.505, Amer. Math. Soc., Providence, RI, 2010, pp. 171-178.DOI: 10.1090/conm/505/09922. · Zbl 1204.42035
[29] M. T. Lacey, S. Petermichl, J. C. Pipher, and B. D. Wick, Multi-parameter Div-Curl lemmas,Bull. Lond. Math. Soc.44(6)(2012), 1123-1131.DOI: 10. 1112/blms/bds037. · Zbl 1257.35058
[30] M. T. Lacey, E. T. Sawyer, C.-Y. Shen, and I. Uriarte-Tuero, Two-weight inequality for the Hilbert transform: a real variable characterization, I,Duke Math. J.163(15)(2014), 2795-2820.DOI: 10.1215/00127094-2826690. · Zbl 1312.42011
[31] A. K. Lerner, S. Ombrosi, and I. P. Rivera-R´ıos, On pointwise and weighted estimates for commutators of Calder´on-Zygmund operators,Adv. Math.319 (2017), 153-181.DOI: 10.1016/j.aim.2017.08.022. · Zbl 1379.42007
[32] A. K. Lerner, S. Ombrosi, and I. P. Rivera-R´ıos, Commutators of singular integrals revisited,Bull. Lond. Math. Soc.51(1)(2019), 107-119.DOI: 10.1112/ blms.12216. · Zbl 1418.42021
[33] K. Li, H. Martikainen, Y. Ou, and E. Vuorinen, Bilinear representation theorem,Trans. Amer. Math. Soc.371(6)(2019), 4193-4214.DOI: 10.1090/ tran/7505. · Zbl 1417.42018
[34] K. Li, H. Martikainen, and E. Vuorinen, Bloom-type inequality for biparameter singular integrals:efficient proof and iterated commutators,Int. Math. Res. Not. IMRN(2019).DOI: 10.1093/imrn/rnz072. · Zbl 1478.42013
[35] K. Li, H. Martikainen, and E. Vuorinen, Bilinear Calder´on-Zygmund theory on product spaces,J. Math. Pures Appl. (9)138(2020), 356-412.DOI: 10.1016/ j.matpur.2019.10.007. · Zbl 1439.42020
[36] K. Li, H. Martikainen, and E. Vuorinen, Bloom type upper bounds in the product BMO setting,J. Geom. Anal.30(2020), 3181-3203.DOI: 10.1007/ s12220-019-00194-3. · Zbl 1440.42063
[37] H. Martikainen, Representation of bi-parameter singular integrals by dyadic operators,Adv. Math.229(3)(2012), 1734-1761.DOI: 10.1016/j.aim.2011.12. 019. · Zbl 1241.42012
[38] F. Nazarov, S. Treil, and A. Volberg, TheT b-theorem on non-homogeneous spaces,Acta Math.190(2)(2003), 151-239.DOI: 10.1007/BF02392690. · Zbl 1065.42014
[39] Y. Ou, Multi-parameter singular integral operators and representation theorem, Rev. Mat. Iberoam.33(1)(2017), 325-350.DOI: 10.4171/RMI/939. · Zbl 1366.42016
[40] Y. Ou, S. Petermichl, and E. Strouse, Higher order Journ´e commutators and characterizations of multi-parameter BMO,Adv. Math.291(2016), 24-58. DOI: 10.1016/j.aim.2015.12.029. · Zbl 1335.42013
[41] S.
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