乘积线性算子的 ${H_{b_1,b_2}^{p,w}(\mathbb{R} ^n\times\mathbb{R}^m)} \to {L^p_w}(\mathbb{R}^{n+m})$ 有界性

数学物理学报 ›› 2026, Vol. 46 ›› Issue (6): 2075-2092.

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乘积线性算子的 ${H_{b_1,b_2}^{p,w}(\mathbb{R} ^n\times\mathbb{R}^m)} \to {L^p_w}(\mathbb{R}^{n+m})$ 有界性

郑涛涛¹^,^*(), 卢嘉谊¹(), 肖燕梅²(), 周需焕³()

¹ 浙江科技大学理学院杭州 310023
² 中国矿业大学理学院北京 100083
³ 南京警察学院基础部南京 210023

收稿日期:2025-06-23 修回日期:2026-01-06 出版日期:2026-12-26 发布日期:2026-08-14
通讯作者: 郑涛涛 E-mail:zhengtao@zust.edu.cn;lllllujy@foxmail.com;yanmei.xiao@foxmail.com;zhouxuhuan@163.com
作者简介:卢嘉谊,E-mail: lllllujy@foxmail.com;
肖燕梅, E-mail: yanmei.xiao@foxmail.com;
周需焕, E-mail: zhouxuhuan@163.com
基金资助:
国家自然科学基金(12326308);浙江科技大学研究生科研创新基金(2024yjskc20);浙江科技大学研究生科研创新基金(2023JLYB011);浙江科技大学研究生科研创新基金(2021yjsjg09);浙江科技大学研究生科研创新基金(JGCG2024340);江苏省自然科学基金青年项目(BK20200587);北京师范大学数学与复杂系统教育部重点实验室开放课题(K202304)

The Boundedness of the Product Linear Operator from ${H_{b_1,b_2}^{p,w}(\mathbb{R} ^n\times\mathbb{R}^m)}$ to ${L^p_w}(\mathbb{R}^{n+m})$

Taotao Zheng¹^,^*(), Jiayi Lu¹(), Yanmei Xiao²(), Xuhuan Zhou³()

¹ School of Science, Zhejiang University of Science and Technology, Hangzhou 310023
² Department of Mathematics, China University of Mining & Technology(Beijing), Beijing 100083
³ Department of Basic Courses, Nanjing Police University, Nanjing 210023

Received:2025-06-23 Revised:2026-01-06 Online:2026-12-26 Published:2026-08-14
Contact: Taotao Zheng E-mail:zhengtao@zust.edu.cn;lllllujy@foxmail.com;yanmei.xiao@foxmail.com;zhouxuhuan@163.com
Supported by:
NSFC(12326308);Postgraduate Research Innovation Fund of Zhejiang University of Science and Technology(2024yjskc20);Postgraduate Research Innovation Fund of Zhejiang University of Science and Technology(2023JLYB011);Postgraduate Research Innovation Fund of Zhejiang University of Science and Technology(2021yjsjg09);Postgraduate Research Innovation Fund of Zhejiang University of Science and Technology(JGCG2024340);Natural Science Foundation of Jiangsu Province(BK20200587);Open Project Program of Key Laboratory of Mathematics and Complex System(K202304);Beijing Normal University

摘要/Abstract

摘要：

该文建立了一类乘积线性算子从与仿增长函数相关的加权乘积 Hardy 空间到加权乘积 Lebesgue 空间上有界的一般性方法. 首先, 利用离散型 Calderón 再生公式建立与仿增长函数 $b_1,b_2$ 相关的加权乘积 Hardy 空间 $H_ {b_1,b_2}^{p,w}(\mathbb{R} ^n\times\mathbb{R}^m)$ 的连续型与离散型的等价刻画, 再结合稠密性命题以及 $f\in L^q(\mathbb{R} ^{n+m})\cap H_{b_1,b_2}^{p,w}(\mathbb{R} ^n\times\mathbb{R}^m)$ 时的 ${H_{b_1,b_2}^{p,w}(\mathbb{R} ^n\times\mathbb{R}^m)}$ 范数与 ${L^p_w}(\mathbb{R}^{n+m})$ 范数控制关系来得到相关结论.作为应用, 得到了广义乘积 Calderón-Zygmund 算子从与仿增长函数相关的乘积 Hardy 空间到乘积 Lebesgue 空间上有界性, 避免了使用与仿增长函数相关的乘积 Hardy 空间的原子分解理论和 Journé 覆盖引理.

关键词: 乘积线性算子, 加权乘积 Hardy 空间, 仿增长函数, Plancherel-P?lya 不等式.

Abstract:

This paper establishes a general method for proving the boundedness of a class of product linear operators from weighted product Hardy spaces associated with para-accretive functions to weighted product Lebesgue spaces. Firstly, by using the discrete Calderón reproducing formula, the equivalent characterizations of both the continuous and discrete forms for the weighted product Hardy spaces ${H_{b_1,b_2}^{p,w}(\mathbb{R} ^n\times\mathbb{R}^m)}$ associated with para-accretive functions $b_1$ and $b_2$ are established. Then, combining the density proposition and the norm domination relationship between the $H_{b_1,b_2}^{p,w}(\mathbb{R} ^n\times\mathbb{R}^m)$ norm and the $ L^p_w(\mathbb{R} ^{n+m})$ norm for $f \in L^q(\mathbb{R} ^{n+m}) \cap H_{b_1,b_2}^{p,w}(\mathbb{R} ^n\times\mathbb{R}^m)$, the main results are obtained. As an application, the boundedness of generalized product Calderón-Zygmund operators from product Hardy spaces associated with para-accretive functions to product Lebesgue spaces is established. This approach avoids using the atomic decomposition theory of product Hardy spaces associated with para-accretive functions and the Journé covering lemma.

Key words: product linear operator, weighted product Hardy space, para-accretive function, Plancherel-P?lya inequality.

中图分类号:

O174.3

郑涛涛, 卢嘉谊, 肖燕梅, 周需焕. 乘积线性算子的 ${H_{b_1,b_2}^{p,w}(\mathbb{R} ^n\times\mathbb{R}^m)} \to {L^p_w}(\mathbb{R}^{n+m})$ 有界性[J]. 数学物理学报, 2026, 46(6): 2075-2092.

Taotao Zheng, Jiayi Lu, Yanmei Xiao, Xuhuan Zhou. The Boundedness of the Product Linear Operator from ${H_{b_1,b_2}^{p,w}(\mathbb{R} ^n\times\mathbb{R}^m)}$ to ${L^p_w}(\mathbb{R}^{n+m})$[J]. Acta mathematica scientia,Series A, 2026, 46(6): 2075-2092.

参考文献 33

[1]	Calderón A P, Zygmund A. On the existence of certain singular integrals. Acta Math, 1952, 88, 85-139 doi: 10.1007/BF02392130
[2]	David G, Journé J L. A boundedness criterion for generalized Calderón-Zygmund operators. Ann of Math, 1984, 120(2): 371-397 doi: 10.2307/2006946
[3]	Mclntosh A, Meyer Y. Algebras of singular integral operators, C R Acad Sci Paris Sér I Math, 1985, 301(8): 395-397
[4]	David G, Journé J L, Semmes S. Calderón-Zygmund operators, para-accretive functions and interpolation. Rev Mat Iberoam, 1985, 1(4): 1-56
[5]	Han Y S, Lee M Y, Lin C C. Para-accretive functions and Hardy spaces. Integr Equ Oper Th, 2012, 72(1): 67-90
[6]	Deng D G, Yang D C. Some new Besov and Triebel-Lizorkin spaces associated with para-accretive functions on spaces of homogeneous type. J Aust Math Soc, 2006, 80(2): 229-262 doi: 10.1017/S1446788700013094
[7]	Liao F H, Liu Z G, Zhang X J. The $Tb$ theorem for inhomogeneous Besov and Triebel-Lizorkin spaces and its application. Georgian Math J, 2016, 23(2): 253-267 doi: 10.1515/gmj-2015-0062
[8]	Zheng T T, Tao X X. $Tb$ criteria for Calderón-Zygmund operators on Lipschitz spaces with para-accretive function. Publ Math Debrecen, 2019, 95(3-4): 487-503 doi: 10.5486/PMD
[9]	Lan S H, Lin C C. Weighted Hardy spaces associated to para-accretive functions. Taiwan J Math, 2010, 14(3B): 1055-1078
[10]	Tan J. Weighted variable Hardy spaces associated with para-accretive functions and boundedness of Calderón-Zygmund operators. J Geom Anal, 2023, 33: 61 doi: 10.1007/s12220-022-01121-9
[11]	Han Y S, Sawyer E T. Para-accretive functions, the weak boundedness property and the $Tb$ theorem. Rev Mat Iberoam, 1990, 6(1-2): 17-41 doi: 10.4171/rmi
[12]	Han Y S. Calderón-type reproducing formula and the $Tb$ theorem. Rev Mat Iberoam, 1994, 10(1): 51-91 doi: 10.4171/rmi
[13]	Tao X X, Kang Y C, Zheng T T. The $Tb$ theorem for some inhomogeneous Besov and Triebel-Lizorkin spaces over space of homogeneous type. J Math Anal Appl, 2024, 531(1): Art 127879
[14]	Han Y S, Lee M Y, Lin C C. Hardy spaces and the $Tb$ theorem. J Geom Anal, 2004, 14(2): 291-318 doi: 10.1007/BF02922074
[15]	Fefferman R, Stein E M. Singular integrals on product spaces. Adv Math, 1982, 45(2): 117-143 doi: 10.1016/S0001-8708(82)80001-7
[16]	Journé J L. Calderón-Zygmund operators on product spaces. Rev Mat Iberoam, 1985, 1(3): 55-91 doi: 10.4171/rmi
[17]	Chang S Y A, Fefferman R. A continuous version of the duality of $H^1$ and BMO on the bidisc. Ann of Math, 1980, 112(1): 179-201 doi: 10.2307/1971324
[18]	Chang S Y A, Fefferman R. Some recent developments in Fourier analysis and $H^p$-theory on product domains. Bull Amer Math Soc (N S), 1985, 12(1): 1-43
[19]	Fefferman R. Harmonic analysis on product spaces. Ann of Math, 1987, 126(1): 109-130 doi: 10.2307/1971346
[20]	Han Y S, Lee M Y, Lin C C, et al. Calderón-Zygmund operators on product Hardy spaces. J Funct Anal, 2010, 258(8): 2834-2861 doi: 10.1016/j.jfa.2009.10.022
[21]	Lee M Y, Li J, Lin C C. Product Hardy spaces associated with para-accretive functions and $Tb$ theorem. New York J Math, 2019, 25: 1438-1484
[22]	Zheng T T, Tao X X. $Tb$ theorem for the generalized singular integral operator on product Lipschitz spaces with para-accretive functions. New York J Math, 2020, 26: 1028-1063
[23]	Bao J Y, Tan J, Zhao J M. Product local Hardy spaces associated with ball quasi-Banach function spaces. J Pseudo-Differ Oper Appl, 2025, 16(1): Art 16
[24]	Chen P, Duong X T, Li J, et al. Product Hardy spaces associated to operators with heat kernel bounds on spaces of homogeneous type. Math Z, 2016, 282(3-4): 1033-1065 doi: 10.1007/s00209-015-1577-6
[25]	Han Y S, Lu G Z. Discrete Littlewood-Paley-Stein theory and multi-parameter Hardy spaces associated with the flag singular integrals. arXiv:0801.1701
[26]	Han Y S, Li J, Lu G Z, et al. $H^p\to H^p$ boundedness implies $H^p\to L^p$ boundedness. Forum Math, 2011, 23(4): 729-756 doi: 10.1515/form.2011.026
[27]	He Z Y, Yan X J, Yang D C. Calderón reproducing formulae on product spaces of homogeneous type and their applications. Math Nachr, 2025, 298(6): 1839-1921 doi: 10.1002/mana.v298.6
[28]	李子燕, 陶祥兴. 基于小波基的乘积加权 Besov 空间和乘积加权 Triebel-Lizorkin 空间的 Plancherel-Pôlya 型刻画. 数学物理学报, 2025, 45A(3): 665-686
	Li Z Y, Tao X X. Plancherel-Pôlya type characterization of product weighted Besov and product weighted Triebel-Lizorkin spaces based on wavelet basis. Acta Math Sci, 2025, 45A(3): 665-686
[29]	Wang W. A discrete transform and Triebel-Lizorkin spaces on the bidisc. Trans Amer Math Soc, 1995, 347(4): 1351-1364 doi: 10.1090/tran/1995-347-04
[30]	Zheng T T, Chen J C, Dai J W, et al. Calderón-Zygmund operators on homogeneous product Lipschitz spaces. J Geom Anal, 2021, 31(2): 2033-2057 doi: 10.1007/s12220-019-00331-y
[31]	Zheng T T, Xiao Y M, He S Y, et al. $T1$ theorem on homogeneous product Besov spaces and product Triebel-Lizorkin spaces. Banach J Math Anal, 2022, 16(3): 1278-1311
[32]	Zheng T T, Xiao Y M, Tao X X. Weighted estimates for product singular integral operators in Journé's class on RD-spaces. Forum Math, 2025, 37(2): 593-627
[33]	郑涛涛, 肖燕梅, 陶祥兴. RD 空间上广义乘积 Calderón-Zygmund 算子在端点函数空间上的 $T1$ 定理. 中国科学: 数学, 2023, 53(3): 441-472
	Zheng T T, Xiao Y M, Tao X X. The $T1$ theorem for the generalized product Calderón-Zygmund operator on product endpoint function spaces over RD spaces. Sci Sin Math, 2023, 53(3): 441-472 doi: 10.1360/SSM-2021-0189

乘积线性算子的 ${H_{b_1,b_2}^{p,w}(\mathbb{R} ^n\times\mathbb{R}^m)} \to {L^p_w}(\mathbb{R}^{n+m})$ 有界性

The Boundedness of the Product Linear Operator from ${H_{b_1,b_2}^{p,w}(\mathbb{R} ^n\times\mathbb{R}^m)}$ to ${L^p_w}(\mathbb{R}^{n+m})$

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