Marcinkiewicz积分算子及其交换子在非齐度量测度空间上的有界性

数学物理学报 ›› 2020, Vol. 40 ›› Issue (3): 597-610.

Marcinkiewicz积分算子及其交换子在非齐度量测度空间上的有界性

韩瑶瑶(),赵凯*()

^{青岛大学数学与统计学院山东青岛 266071}

收稿日期:2019-08-27 出版日期:2020-06-26 发布日期:2020-07-15
通讯作者: 赵凯 E-mail:hanyaoyao@aliyun.com;zhkzhc@aliyun.com
作者简介:韩瑶瑶, E-mail:hanyaoyao@aliyun.com
基金资助:
国家自然科学基金(11471176);国家自然科学基金(11871293)

Boundedness of Marcinkiewicz Integral and Its Commutator on Non-Homogeneous Metric Measure Spaces

Yaoyao Han(),Kai Zhao*()

^{School of Mathematics and Statistics, Qingdao University, Shandong Qingdao 266071}

Received:2019-08-27 Online:2020-06-26 Published:2020-07-15
Contact: Kai Zhao E-mail:hanyaoyao@aliyun.com;zhkzhc@aliyun.com
Supported by:
the NSFC(11471176);the NSFC(11871293)

摘要/Abstract

摘要：

设$ ({\cal X}, d, \mu) $是一个满足上双倍条件和几何双倍条件的非齐度量测度空间.利用非齐度量测度空间的一些特征和不等式技巧，证明了Marcinkiewicz积分算子及其交换子在非齐度量测度空间上的Herz空间以及Herz型Hardy空间上的有界性.

关键词: 非齐度量测度空间, Herz空间, Hardy空间, Marcinkiewicz积分算子, 有界性

Abstract:

Let $ ({\cal X}, d, \mu) $ be a non-homogeneous metric measure space satisfying both the geometrically doubling and the upper doubling conditions. By using the properties of non-homogeneous metric measure space and inequality technique, the authors proved that the Marcinkiewicz integral operator and its commutator are bounded on Herz spaces and Herz type Hardy spaces with non-homogeneous metric measure space.

Key words: Non-homogeneous metric measure space, Herz space, Hardy space, Marcinkiewicz integral, Boundedness

中图分类号:

O174.2

韩瑶瑶,赵凯. Marcinkiewicz积分算子及其交换子在非齐度量测度空间上的有界性[J]. 数学物理学报, 2020, 40(3): 597-610.

Yaoyao Han,Kai Zhao. Boundedness of Marcinkiewicz Integral and Its Commutator on Non-Homogeneous Metric Measure Spaces[J]. Acta mathematica scientia,Series A, 2020, 40(3): 597-610.

参考文献

1	Coifman R R , Weiss G . Analyse Harmonique Non-Commutative sur Certains Espaces Homogénes. Berlin: Springer-Verlag, 1971
2	Tolsa X . BMO, H¹ and Calderón-Zygmund operators for non-doubling measures. Math Ann, 2001, 319 (1): 89- 149
3	Hu G E , Meng Y , Yang D C . Endpoint estimate for maxmial commutators with non-doubling measures. Acta Math Sci, 2006, 26B (2): 271- 280
4	Lin H B , Meng Y , Yang D C . Weighted estimates for commutators of multilinear Calderón-Zygmund operators with non-doubling measures. Acta Math Sci, 2010, 30B (1): 1- 18
5	Yang D C , Yang D Y , Hu G E . The Hardy Space H¹ with Non-Doubling Measures and Their Application. Berlin: Springer-Verlag, 2013
6	Hyt?nen T . A framework for non-homogeneous analysis on metric spaces, and the RBMO space of Tolsa. Publ Mat, 2010, 54, 485- 504
7	Fu X , Lin H B , Yang D C , et al. Hardy spaces H^p over non-homogeneous metric measure spaces and their applications. Sci China Math, 2015, 58, 309- 388
8	韩瑶瑶, 赵凯. 非齐度量测度空间上的Herz型Hardy空间. 中国科学:数学, 2018, 48 (10): 1315- 1338
8	Han Y Y , Zhao K . Herz type Hardy spaces on non-homogeneous metric measure spaces. Sci Sin Math, 2018, 48 (10): 1315- 1338
9	Stein E M . On the functions of Littlewood-Paley, Lusin and Marcinkiewicz. Trans Amer Math Soc, 1958, 88 (2): 430- 466
10	Ding Y , Fan D S , Pan Y B . Weighted boundedness for a class of rough Marcinkiewicz integrals. Indiana Univ Math J, 1999, 48 (3): 1037- 1055
11	Chen D X , Chen J C . Boundedness of Marcinkiewicz integrals with rough kernels on Herz spaces. Adv Math (China), 2005, 34 (5): 591- 599
12	Mo H X , Lu S Z . Boundedness of generalized high commutators of Marcinkiewicz integrals. Acta Math Sci, 2007, 27B (4): 852- 866
13	程美芳, 束立生. Marcinkiewicz积分算子交换子在Hardy空间及Herz型Hardy空间上的有界性. 数学物理学报, 2008, 28A (2): 222- 231
13	Cheng M F , Shu L S . Boundedness of the commutator of Marcinkiewicz integral on Hardy space and Herz-type Hardy space. Acta Math Sci, 2008, 28A (2): 222- 231
14	陶祥兴, 位瑞英. 可变核Marcinkiewicz积分交换子在Herz型Hardy空间上的有界性. 数学物理学报, 2009, 29A (6): 1508- 1517
14	Tao X X , Wei R Y . Boundedness of commutators related to Marcinkiewicz integrals with variable kernels in Herz-type Hardy spaces. Acta Math Sci, 2009, 29A (6): 1508- 1517
15	王洪彬, 傅尊伟, 刘宗光. 变指标Lebesgue空间上的Marcinkiewicz积分高阶交换子. 数学物理学报, 2012, 32A (6): 1092- 1101
15	Wang H B , Fu Z W , Liu Z G . Higher-order commutators of Marcinkiewicz integrals on variable Lebesgue spaces. Acta Math Sci, 2012, 32A (6): 1092- 1101
16	Heinonen J . Lectures on Analysis on Metric Spaces. New York: Springer-Verlag, 2001
17	Hyt?nen T , Yang D C , Yang D Y . The Hardy space H¹ on non-homogeneous metric spaces. Math Proc Cambridge Philos Soc, 2012, 153, 9- 31
18	王萍.非齐度量测度空间上的几类算子[D].兰州:西北师范大学, 2016
18	Wang P. Some Operators on Non-Homogeneous Metric Measure Spaces[D]. Lanzhou: Northwest Normal University, 2016
19	Lin H B , Yang D C . Equivalent boundedness of Marcinkiewicz integrals on non-homogeneous metric measure spaces. Sci China Math, 2014, 57, 123- 144

[1]	邓凯鑫, 梁玉霞, 杨子聪. 调和 Fock 空间上的广义 Volterra 型积分算子研究[J]. 数学物理学报, 2025, 45(5): 1373-1380.
[2]	董建祥. 向量值指数型权 Bergman 空间上的 Hankel 算子[J]. 数学物理学报, 2024, 44(3): 513-524.
[3]	陈建华, 彭建文. 非凸多分块优化的 Bregman ADMM 的收敛率研究[J]. 数学物理学报, 2024, 44(1): 195-208.
[4]	陈洪欣,张学军,周敏. $ \boldsymbol{H^{p,q,s}}({\Bbb D})$上的复合算子[J]. 数学物理学报, 2023, 43(1): 1-13.
[5]	丁宣浩, 黄雨浩, 李永宁. 调和Hardy空间上的Toeplitz算子的酉等价性[J]. 数学物理学报, 2023, 43(1): 27-34.
[6]	张传洲,王超越,张学英. Hardy空间上维林肯型系统下的极大算子[J]. 数学物理学报, 2022, 42(5): 1294-1305.
[7]	郭雨婷,张学军. 正规权一般函数空间到Bloch型空间的复合算子[J]. 数学物理学报, 2022, 42(5): 1306-1319.
[8]	徐宁,周泽华,丁颖. 单位球Hardy空间上加权复合算子的交换子[J]. 数学物理学报, 2021, 41(6): 1606-1615.
[9]	贾哲,杨作东. 带非线性扩散项和信号产生项的趋化-趋触模型解的整体有界性[J]. 数学物理学报, 2021, 41(5): 1382-1395.
[10]	温泳铭,侯宪明. 拟微分算子在$H^p(\omega)$上的有界性[J]. 数学物理学报, 2021, 41(2): 303-312.
[11]	陈萱. Heisenberg群上与Schrödinger算子相关的Riesz变换在Hardy空间上的有界性[J]. 数学物理学报, 2021, 41(1): 46-62.
[12]	逯光辉,陶双平. 度量测度空间上θ型Marcinkiewicz积分及交换子[J]. 数学物理学报, 2020, 40(6): 1431-1445.
[13]	阳卫锋. 复微分方程解的增长性及解在若干函数空间的条件[J]. 数学物理学报, 2020, 40(6): 1481-1491.
[14]	唐鹏程,徐思,张学军. Cⁿ中对数权一般函数空间上的Bergman型算子[J]. 数学物理学报, 2020, 40(5): 1151-1162.
[15]	赵艳辉,吴修云,廖春艳. 单位球上加权Bergman空间到Ƶ_μ型空间的加权Cesàro算子[J]. 数学物理学报, 2020, 40(3): 569-578.