$L^P$ BOUNDEDNESS OF COMMUTATOR AND HÖRMANDER CONDITION

doi:10.1007/s10473-025-0519-6

数学物理学报(英文版) ›› 2025, Vol. 45 ›› Issue (5): 2171-2189.doi: 10.1007/s10473-025-0519-6

$L^P$ BOUNDEDNESS OF COMMUTATOR AND HÖRMANDER CONDITION

Qixiang YANG¹, Haibo YANG², Chitin HON², Tao QIAN^3,*

1. School of Mathematics and Statistics, Wuhan University, Wuhan 430071, China;
2. Macau Institute of Systems Engineering, Macau University of Science and Technology, Macau 999078, China;
3. Director of Macau Centre of of Mathematical Sciences, Faculty of Innovation Engineering, Macau University of Science and Technology, Macau 999078, China

收稿日期:2024-03-29 修回日期:2024-10-22 出版日期:2025-09-25 发布日期:2025-10-14

$L^P$ BOUNDEDNESS OF COMMUTATOR AND HÖRMANDER CONDITION

Qixiang YANG¹, Haibo YANG², Chitin HON², Tao QIAN^3,*

1. School of Mathematics and Statistics, Wuhan University, Wuhan 430071, China;
2. Macau Institute of Systems Engineering, Macau University of Science and Technology, Macau 999078, China;
3. Director of Macau Centre of of Mathematical Sciences, Faculty of Innovation Engineering, Macau University of Science and Technology, Macau 999078, China

Received:2024-03-29 Revised:2024-10-22 Online:2025-09-25 Published:2025-10-14
Contact: ^*Tao Qian, E-mail: tqian@must.edu.mo
About author:Qixiang Yang, E-mail: qxyang@whu.edu.cn; Haibo Yang, E-mail: yanghb97@qq.com; Chitin Hon, E-mail: cthon@must.edu.mo
Supported by:
This research was partially supported by the research grant of Macau University of Science and Technology (FRG-22-075-MCMS), the Macau Government Research Funding (FDCT0128/2022/A), the Science and Technology Development Fund of Macau SAR (005/2022/ALC), the Science and Technology Development Fund of Macau SAR (0045/2021/A), Macau University of Science and Technology (FRG-20-021-MISE).

摘要/Abstract

摘要： For $1<p<\infty$, Coifman-Rochberg-Weiss established $L^{p}$ boundedness of commutators of smooth kernels. Later, many works tried to weaken the smooth condition. In this paper, we extend these mentioned results to the case of non-homogeneous but with strong Hörmander condition. Our main skills lie in wavelet decomposition, wavelet commutators, Hardy-Littlewood maximal operator and Fefferman-Stein's vector-valued maximum function Theorem.

关键词: wavelets, commutator, Hardy space, Hörmander condition, vector-valued maximum function, $L^p$ boundedness

Abstract: For $1<p<\infty$, Coifman-Rochberg-Weiss established $L^{p}$ boundedness of commutators of smooth kernels. Later, many works tried to weaken the smooth condition. In this paper, we extend these mentioned results to the case of non-homogeneous but with strong Hörmander condition. Our main skills lie in wavelet decomposition, wavelet commutators, Hardy-Littlewood maximal operator and Fefferman-Stein's vector-valued maximum function Theorem.

Key words: wavelets, commutator, Hardy space, Hörmander condition, vector-valued maximum function, $L^p$ boundedness

中图分类号:

32A50

Qixiang YANG, Haibo YANG, Chitin HON, Tao QIAN. $L^P$ BOUNDEDNESS OF COMMUTATOR AND HÖRMANDER CONDITION[J]. 数学物理学报(英文版), 2025, 45(5): 2171-2189.

Qixiang YANG, Haibo YANG, Chitin HON, Tao QIAN. $L^P$ BOUNDEDNESS OF COMMUTATOR AND HÖRMANDER CONDITION[J]. Acta mathematica scientia,Series B, 2025, 45(5): 2171-2189.

参考文献

[1] Chen D X, Lu S Z. $L^p$ boundedness for parabolic Littlewood-Paley operator with rough kernel belonging to $F(S^{(n-1)})$. Acta Math Sci, 2011, 31: 343-350
[2] Cheng M D, Deng D G, Long R L. Real Analysis.Beijing: High Education Press, 2008
[3] Coifman R, Rochberg R, Weiss G. Factorization theorems for Hardy spaces in several variables. Ann Math, 1976, 103: 611-635
[4] Connett W C.Singular integrals near $L^1$//Weiss G, Wainger S. Harmonic Analysis in Euclidean Spaces. Providence, RI: American Mathematical Society, 1979
[5] Deng D G, Yan L X. $L^p$ boundedness of commutators of Calderón-Zygmund singular integral operators, Adv Math (China), 1998, 3(3): 259-269
[6] Deng D G, Yan L X, Yang Q X. $L^{2}$ boundedness of commutators of Calderón-Zygmundsingular integral operators. Progress in Natural Science, 1998, 8(4): 416-427
[7] Fefferman C, Stein E M. Some maximal inequalities. Amer Math, 1971, 93: 107-115
[8] Grafakos L, Stefanov A.Convolution Calderón-Zygmund singular integral operators with rough kernels// Bray W O, Stanojević $\check{\rm C}$ V. Analysis of Divergence. Boston, MA: Birkhäuser Boston, 1999
[9] Guo W, He J, Wu H. Limiting weak-type behaviors for certain classical operators in harmonic analysis. Potential Anal, 2021, 54(2): 307-330
[10] Guo X, Hu G. On the commutators of singular integral operators with rough convolution kernels. Canad J Math, 2016, 68(4): 816-840
[11] Meyer Y.Ondelettes et opérateurs, I et II. Paris: Hermann, 1991
[12] Meyer Y, Yang Q X. Continuity of Calderón-Zygmund operators on Besov or Triebel-Lizorkin spaces. Anal Appl, 2008, 6(1): 51-81
[13] Triebel H.Theory of Function Spaces. Basel: Birkhäuser-Verlag, 1983
[14] Yang Q X, Ding Y. Fast algorithm for Calderón-Zygmund operators: convergence speed and rough kernel. Acta Math Sci, 2016, 36(2): 345-359
[15] Yang Q X, Lou Z J. Commutators and rough kernels without zero homogeneous condition. Inter J Wavelets Multiresolu Infor Proc, 2018, 16(5): 289-300

$L^P$ BOUNDEDNESS OF COMMUTATOR AND HÖRMANDER CONDITION

$L^P$ BOUNDEDNESS OF COMMUTATOR AND HÖRMANDER CONDITION

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