与算子相连的乘积 Hardy 空间的分子分解——献给邓引斌教授 70 寿辰

数学物理学报 ›› 2026, Vol. 46 ›› Issue (4): 1406-1419.

与算子相连的乘积 Hardy 空间的分子分解——献给邓引斌教授 70 寿辰

陈彧(), 邓清泉^*()

华中师范大学数学与统计学学院武汉 430079

收稿日期:2025-12-25 修回日期:2026-02-10 出版日期:2026-08-26 发布日期:2026-06-10
通讯作者: 邓清泉 E-mail:13237212212@163.com;dengq@mail.ccnu.edu.cn
作者简介:陈彧,E-mail: 13237212212@163.com
基金资助:
国家自然科学基金(12471092);国家自然科学基金(12531005)

The Molecular Characterization for Product Hardy Space Associated to Operators on Product Domain

Yu Chen(), Qingquan Deng^*()

School of Mathematics and Statistics, Central China Normal University, Wuhan 430079

Received:2025-12-25 Revised:2026-02-10 Online:2026-08-26 Published:2026-06-10
Contact: Qingquan Deng E-mail:13237212212@163.com;dengq@mail.ccnu.edu.cn
Supported by:
NSFC(12471092);NSFC(12531005)

摘要/Abstract

摘要：

在该文中, 假设非负自伴算子$ L_{1}$ 与$ L_{2}$ 的热半群满足非对角估计$ (GGE_{p_{0}})$, 其中$ p_{0}\in[1,2)$. 在乘积空间$ \mathbb{R}^{n_{1}}\times \mathbb{R}^{n_{2}}$ 中, 通过自伴算子$ L_{1}$ 与$ L_{2}$ 的热半群生成的面积积分定义了与算子相连的 Hardy 空间$ H_{L_1, L_2}^{1}(\mathbb{R}^{n_1}\times \mathbb{R}^{n_2})$, 借助乘积帐篷空间理论, 给出了乘积 Hardy 空间的分子分解理论.

关键词: 面积积分, 乘积 Hardy 空间, 分子分解

Abstract:

In this paper, assume that $L_{1}$ and $L_{2}$ be self-adjoint operators, and the corresponding heat semigroups ${\rm e}^{-tL_{1}}$ and ${\rm e}^{-tL_{2}}$ satisfy off-diagonal estimates of type ($GGE_{p_{0},m}$) for some $p_{0}\in [1,2)$, we introduce the Hardy space $H_{L_{1}, L_{2}}(\mathbb{R}^{n_{1}}\times \mathbb{R}^{n_{2}})$ associated to $L_{1}$ and $L_{2}$ in terms of the area function, and establish the molecular decomposition by using the theory of tent space on product domain.

Key words: area function, product Hardy space, molecular decomposition

中图分类号:

O174.2

陈彧, 邓清泉. 与算子相连的乘积 Hardy 空间的分子分解——献给邓引斌教授 70 寿辰[J]. 数学物理学报, 2026, 46(4): 1406-1419.

Yu Chen, Qingquan Deng. The Molecular Characterization for Product Hardy Space Associated to Operators on Product Domain[J]. Acta mathematica scientia,Series A, 2026, 46(4): 1406-1419.

参考文献 44

[1]	Auscher P. On Necessary and Sufficient Conditions for $L^{p}$-estimates of Riesz Transforms Associated to Elliptic Operators on $\mathbb{R}^{n}$ and Related Estimates. Providence RI: Mem Amer Math Soc, 2007
[2]	Auscher P, Duong X T, Mcintosh A. Boundedness of Banach space valued singular integral operators and Hardy spaces. unpublished preprint, 2005, 3(5): Art 4
[3]	Cao J, Yang D. Hardy spaces $H_L^p(\mathbb{R}^n)$ associated with operators satisfying $k$-Davies-Gaffney estimates. Sci China Math, 2012, 55: 1403-1440 doi: 10.1007/s11425-012-4394-y
[4]	Chang S Y, R. Fefferman R. The Calderón-Zygmund decomposition on product domains. Amer J Math, 1982, 104: 445-468
[5]	Chen P, Duong X T, Li J, et al. Marcinkiewicz-type spectral multipliers on Hardy and Lebesgue spaces on product spaces of homogeneous type. J Fourier Anal Appl, 2017, 23: 21-64 doi: 10.1007/s00041-016-9460-3
[6]	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: 1033-1065 doi: 10.1007/s00209-015-1577-6
[7]	Chang D C, Yang D, Zhou Y. Boundedness of sublinear operators on product Hardy spaces and its application. J Math Soc Japan, 2010, 62: 321-353
[8]	Coifman R. A real variable characterization of $H^{p}$. Studia Math, 1974, 51: 269-274 doi: 10.4064/sm-51-3-269-274
[9]	Coifman R, Weiss G. Extensions of Hardy spaces and their use on analysis. Bull Amer Math Soc, 1977, 83: 569-645 doi: 10.1090/bull/1977-83-04
[10]	Coifman R, Meyer Y, Stein E M. Some new function spaces and their applications to harmonic analysis. J of funct Anal, 1985, 62(2): 304-335 doi: 10.1016/0022-1236(85)90007-2
[11]	Deng D, Song L, Tan C, et al. Duality of Hardy and BMO spaces associared with operators with heat kernel bounds on product domains. J Geom Anal, 2007, 17: 455-483 doi: 10.1007/BF02922092
[12]	Deng Q, Ding Y, Yao X. Characterizations of Hardy spaces associated to higher order elliptic oprators. J Funct Anal, 2012, 263: 604-674 doi: 10.1016/j.jfa.2012.05.001
[13]	Deng Q, Ding Y, Yao X. Hardy spaces $H^{p}_{L}(\mathbb{R}^{n})$ associated with higher-order Schrödinger type operators. Anal Theory Appl, 2015, 31: 184-206
[14]	Deng Q, Ding Y, Yao X. Riesz transform associated with higher-order Schrödinger type operators. Potential analysis, 2018, 49: 381-410 doi: 10.1007/s11118-017-9661-7
[15]	Deng Q, Guedjiba D. Weighted product Hardy space associated with operators. Front Math China, 2020, 15: 649-683 doi: 10.1007/s11464-020-0852-y
[16]	Deng Q, Guedjiba D. An atomic characterization for product Hardy spaces. Banach J Math Anal, 2026, 20(1): Art 11
[17]	Duong X T, Li J. Hardy spaces associated to operators satisfying Davies-Gaffney estimates and bounded holomorphic functional calculus. J Funct Anal, 2013, 246: 1409-1437
[18]	Duong X T, Li J, Yan L X. Endpoint estimates for singular integrals with non-smooth kernel on product spaces. arXiv:1509.07548
[19]	Duong X T, Li J, Wick B, et al. Characterizations of product Hardy spaces in Bessel setting. J Fourier Anal Appl, 2021, 27(2): Art 24
[20]	Duong X T, Yan L X. New function spaces of BMO type, the John-Nirenberg inequality, interpolation and applications. Comm Pure Appl Math, 2005, 58: 1375-1420 doi: 10.1002/cpa.v58:10
[21]	Duong X T, Yan L X. Duality of Hardy and BMO spaces associated with operators with heat kernel bounds. J Amer Math Soc, 2005, 18: 943-973 doi: 10.1090/jams/2005-18-04
[22]	Fefferman R. Harmonic analysis on product spaces. Ann of Math, 1987, 126: 109-130 doi: 10.2307/1971346
[23]	Fefferman C, Stein E M. $H^p$ spaces of several variables. Acta Math, 1972, 129: 137-193 doi: 10.1007/BF02392215
[24]	Fefferman R, Stein E M. Singular integrals on product spaces. Adv math, 1982, 45: 117-143 doi: 10.1016/S0001-8708(82)80001-7
[25]	Grafakos L. Classical and Modern Fourier Analysis. New York: Springer, 2008
[26]	Gundy R, Stien E M. $H^{p}$ theory for the poly-disc. Proc Natl Acad Sci USA, 1979, 76: 1026-1029 pmid: 16592628
[27]	Hofmann S, Mayboroda S, McIntosh A. Second order elliptic operators with complex bounded measurable coefficients in $L^p$, Sobolev and Hardy spaces. Annales scientifiques de l'Ecole normale supérieure. 2011, 44(5): 723-800
[28]	Han Y, Li J, Lin C. Criterions of the $L^{2}$ boundedness and sharp endpoint estimates for singular integral operators on product spaces of homogeneous type. Ann Sc Norm Super Pisa Cl Sci, 2016, 16: 845-907
[29]	Han Y, Li J, Lu G. Multiparameter Hardy spaces theory on Carnot-Caratheodory spaces and product spaces of homogeneous type. Trans Amer Math Soc, 2013, 365: 319-360 doi: 10.1090/tran/2013-365-01
[30]	Hofmann S, Lu G, Mitrea D, et al. Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates. Memoris of the Amer Math Soc, 2011, 214: Art 1007
[31]	Hofmann S, Mayboroda S. Hardy and BMO spaces associated to divergence form elliptic operators. Math Ann, 2009, 344: 37-116 doi: 10.1007/s00208-008-0295-3
[32]	Jiang R, Yang D. New Orlicz-Hardy spaces associated with divergence form elliptic operators. J Funct Anal, 2010, 258: 1167-1224 doi: 10.1016/j.jfa.2009.10.018
[33]	Jiang R, Yang D, Zhou Y. Orlicz-Hardy spaces associated with operators. Sci China Math, 2009, 52: 1042-1080 doi: 10.1007/s11425-008-0136-6
[34]	Latter R. A characterization of $H^{p}$ in terms of atoms. Studia Math, 1978, 62: 93-101 doi: 10.4064/sm-62-1-93-101
[35]	Li B, Bownik M, Yang D, et al. Weighted anisotropic product Hardy spaces and boundedness of sublinear operators. Math Nachr, 2010, 283: 392-442 doi: 10.1002/mana.v283:3
[36]	Li B, Bownik M, Yang D, et al. Anisotropic singular integrals in product spaces. Sci China Math, 2010, 53: 3163-3178 doi: 10.1007/s11425-010-4108-2
[37]	Li B, Bownik M, Yang D. Littlewood-Paley characterization and duality of weighted anisotropic product Hardy spaces. J Funct Anal, 2014, 266: 2611-2661 doi: 10.1016/j.jfa.2013.12.017
[38]	Uhl M. Spectral multiplier theorems of Hörmander type via generalized Gaussian estimates. Karlsruhe: Karlsruhe Institute of Technology, 2011
[39]	Song L, Tan C. Hardy spaces associated to Schrödinger operators on product spaces. J Funct Spaces, 2012, 179015
[40]	Stein E M, Weiss G. On the theory of harmonic functions of several variables I. The theory of $H^{p}$ spaces. Acta Math, 1960, 103: 25-62 doi: 10.1007/BF02546524
[41]	Taibleson M, Weiss G. Representation Theorems for holomorphic and harmonic functions in $L^{p}$/The Molecular Characterization of Certain Hardy Spaces. Astérisque, 1980, 77: 67-149
[42]	Wilson J. On the atomic decomposition for Hardy spaces. Pacific J Math, 1985, 116: 201-207 doi: 10.2140/pjm
[43]	Yang D, Yang S. Musielak-Orlicz-Hardy spaces associated with operators and their applications. J Geom Anal, 2014, 24: 495-570 doi: 10.1007/s12220-012-9344-y
[44]	Zhao K, Han Y. Boundedness of operators on Hardy spaces. Taiwanese J Math, 2010, 14: 319-327