Zygmund dilations: bilinear analysis and commutator estimates

Airta, Emil; Li, Kangwei; Martikainen, Henri

doi:10.1090/tran/9366

Zygmund dilations: bilinear analysis and commutator estimates
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by Emil Airta, Kangwei Li and Henri Martikainen;

Trans. Amer. Math. Soc.

DOI: https://doi.org/10.1090/tran/9366

Published electronically: April 4, 2025

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Abstract:

We develop both bilinear theory and commutator estimates in the context of entangled dilations, specifically Zygmund dilations $(x_1, x_2, x_3) \mapsto (\delta _1 x_1, \delta _2 x_2, \delta _1 \delta _2 x_3)$ in $\mathbb {R}^3$. We construct bilinear versions of recent dyadic multiresolution methods for Zygmund dilations and apply them to prove a paraproduct free $T1$ theorem for bilinear singular integrals invariant under Zygmund dilations. Independently, we prove linear commutator estimates even when the underlying singular integrals do not satisfy weighted estimates with Zygmund weights. This requires new paraproduct estimates.

References

Emil Airta, Henri Martikainen, and Emil Vuorinen, Product space singular integrals with mild kernel regularity, J. Geom. Anal. 32 (2022), no. 1, Paper No. 24, 49. MR 4349928, DOI 10.1007/s12220-021-00757-3
R. R. Coifman, R. Rochberg, and Guido Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. (2) 103 (1976), no. 3, 611–635. MR 412721, DOI 10.2307/1970954
D. Cruz-Uribe, J. M. Martell, and C. Pérez, Extrapolation from $A_\infty$ weights and applications, J. Funct. Anal. 213 (2004), no. 2, 412–439. MR 2078632, DOI 10.1016/j.jfa.2003.09.002
Javier Duoandikoetxea, Extrapolation of weights revisited: new proofs and sharp bounds, J. Funct. Anal. 260 (2011), no. 6, 1886–1901. MR 2754896, DOI 10.1016/j.jfa.2010.12.015
Xuan Thinh Duong, Ji Li, Yumeng Ou, Jill Pipher, and Brett Wick, Weighted estimates of singular integrals and commutators in the Zygmund dilation setting, arXiv:1905.00999 (2019).
R. Fefferman and J. Pipher, Multiparameter operators and sharp weighted inequalities, Amer. J. Math. 119 (1997), no. 2, 337–369. MR 1439553, DOI 10.1353/ajm.1997.0011
Loukas Grafakos and José María Martell, Extrapolation of weighted norm inequalities for multivariable operators and applications, J. Geom. Anal. 14 (2004), no. 1, 19–46. MR 2030573, DOI 10.1007/BF02921864
Loukas Grafakos and Seungly Oh, The Kato-Ponce inequality, Comm. Partial Differential Equations 39 (2014), no. 6, 1128–1157. MR 3200091, DOI 10.1080/03605302.2013.822885
Loukas Grafakos and Rodolfo H. Torres, Multilinear Calderón-Zygmund theory, Adv. Math. 165 (2002), no. 1, 124–164. MR 1880324, DOI 10.1006/aima.2001.2028
Ana Grau de la Herrán and Tuomas Hytönen, Dyadic representation and boundedness of nonhomogeneous Calderón-Zygmund operators with mild kernel regularity, Michigan Math. J. 67 (2018), no. 4, 757–786. MR 3877436, DOI 10.1307/mmj/1531447374
Yongsheng Han, Ji Li, Chin-Cheng Lin, and Chaoqiang Tan, Singular integrals associated with Zygmund dilations, J. Geom. Anal. 29 (2019), no. 3, 2410–2455. MR 3969431, DOI 10.1007/s12220-018-0081-8
Irina Holmes, Stefanie Petermichl, and Brett D. Wick, Weighted little bmo and two-weight inequalities for Journé commutators, Anal. PDE 11 (2018), no. 7, 1693–1740. MR 3810470, DOI 10.2140/apde.2018.11.1693
Tuomas P. Hytönen, The sharp weighted bound for general Calderón-Zygmund operators, Ann. of Math. (2) 175 (2012), no. 3, 1473–1506. MR 2912709, DOI 10.4007/annals.2012.175.3.9
Tuomas Hytönen, Kangwei Li, Henri Martikainen, and Emil Vuorinen, Multiresolution analysis and Zygmund dilations, arXiv:2203.15777 (2022).
Tosio Kato and Gustavo Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math. 41 (1988), no. 7, 891–907. MR 951744, DOI 10.1002/cpa.3160410704
Andrei K. Lerner, Sheldy Ombrosi, Carlos Pérez, Rodolfo H. Torres, and Rodrigo Trujillo-González, New maximal functions and multiple weights for the multilinear Calderón-Zygmund theory, Adv. Math. 220 (2009), no. 4, 1222–1264. MR 2483720, DOI 10.1016/j.aim.2008.10.014
Kangwei Li, Henri Martikainen, Yumeng Ou, and Emil Vuorinen, Bilinear representation theorem, Trans. Amer. Math. Soc. 371 (2019), no. 6, 4193–4214. MR 3917220, DOI 10.1090/tran/7505
Kangwei Li, Henri Martikainen, and Emil Vuorinen, Bilinear Calderón-Zygmund theory on product spaces, J. Math. Pures Appl. (9) 138 (2020), 356–412 (English, with English and French summaries). MR 4098772, DOI 10.1016/j.matpur.2019.10.007
Kangwei Li, Henri Martikainen, and Emil Vuorinen, Genuinely multilinear weighted estimates for singular integrals in product spaces, Adv. Math. 393 (2021), Paper No. 108099, 49. MR 4340231, DOI 10.1016/j.aim.2021.108099
Henri Martikainen, Representation of bi-parameter singular integrals by dyadic operators, Adv. Math. 229 (2012), no. 3, 1734–1761. MR 2871155, DOI 10.1016/j.aim.2011.12.019
Detlef Müller, Fulvio Ricci, and Elias M. Stein, Marcinkiewicz multipliers and multi-parameter structure on Heisenberg (-type) groups. I, Invent. Math. 119 (1995), no. 2, 199–233. MR 1312498, DOI 10.1007/BF01245180
Alexander Nagel and Stephen Wainger, $L^{2}$ boundedness of Hilbert transforms along surfaces and convolution operators homogeneous with respect to a multiple parameter group, Amer. J. Math. 99 (1977), no. 4, 761–785. MR 450901, DOI 10.2307/2373864
F. Nazarov, S. Treil, and A. Volberg, The $Tb$-theorem on non-homogeneous spaces, Acta Math. 190 (2003), no. 2, 151–239. MR 1998349, DOI 10.1007/BF02392690
F. Ricci and E. M. Stein, Multiparameter singular integrals and maximal functions, Ann. Inst. Fourier (Grenoble) 42 (1992), no. 3, 637–670 (English, with English and French summaries). MR 1182643, DOI 10.5802/aif.1304

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Zygmund dilations: bilinear analysis and commutator estimates
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Abstract: