LIMITING WEAK-TYPE BEHAVIORS FOR CERTAIN LITTLEWOOD-PALEY FUNCTIONS

doi:10.1007/s10473-019-0102-0

数学物理学报(英文版) ›› 2019, Vol. 39 ›› Issue (1): 11-25.doi: 10.1007/s10473-019-0102-0

LIMITING WEAK-TYPE BEHAVIORS FOR CERTAIN LITTLEWOOD-PALEY FUNCTIONS

侯宪明, 伍火熊

School of Mathematical Sciences, Xiamen University, Xiamen 361005, China

收稿日期:2018-01-29 修回日期:2018-05-09 出版日期:2019-02-25 发布日期:2019-03-13
通讯作者: Huoxiong WU E-mail:huoxwu@xmu.edu.cn
作者简介:Xianming HOU,houxianming37@163.com
基金资助:
The research was supported by the NSFC (11771358, 11471041).

LIMITING WEAK-TYPE BEHAVIORS FOR CERTAIN LITTLEWOOD-PALEY FUNCTIONS

Xianming HOU, Huoxiong WU

School of Mathematical Sciences, Xiamen University, Xiamen 361005, China

Received:2018-01-29 Revised:2018-05-09 Online:2019-02-25 Published:2019-03-13
Contact: Huoxiong WU E-mail:huoxwu@xmu.edu.cn
Supported by:
The research was supported by the NSFC (11771358, 11471041).

摘要/Abstract

摘要： In this paper, we establish the following limiting weak-type behaviors of Littlewood Paley g-function g^φ:for nonnegative function f ∈ L¹(Rⁿ),

where f_t(x)=t^-nf(t^-1x) for t > 0. Meanwhile, the corresponding results for Marcinkiewicz integral and its fractional version with kernels satisfying L_α^q-Dini condition are also given.

关键词: limiting behaviors, weak-type bounds, Littlewood-Paley g-functions, Marcinkiewicz integrals, L_α^q-Dini conditions

Abstract: In this paper, we establish the following limiting weak-type behaviors of Littlewood Paley g-function g^φ:for nonnegative function f ∈ L¹(Rⁿ),

where f_t(x)=t^-nf(t^-1x) for t > 0. Meanwhile, the corresponding results for Marcinkiewicz integral and its fractional version with kernels satisfying L_α^q-Dini condition are also given.

Key words: limiting behaviors, weak-type bounds, Littlewood-Paley g-functions, Marcinkiewicz integrals, L_α^q-Dini conditions

侯宪明, 伍火熊. LIMITING WEAK-TYPE BEHAVIORS FOR CERTAIN LITTLEWOOD-PALEY FUNCTIONS[J]. 数学物理学报(英文版), 2019, 39(1): 11-25.

Xianming HOU, Huoxiong WU. LIMITING WEAK-TYPE BEHAVIORS FOR CERTAIN LITTLEWOOD-PALEY FUNCTIONS[J]. Acta mathematica scientia,Series B, 2019, 39(1): 11-25.

参考文献

[1] Al-Qassem H, Cheng L, Pan Y. On generalized Littlewood-Paley functions. Collect Math, 2018, 69(2):297-314
[2] Benedek A, Calderón A, Panzone R. Convolution operators on Banach space valued functions. Proc Natl Acad Sci USA, 1962, 48(3):356-365
[3] Ding Y, Lai X. L1-Dini conditions and limiting behavior of weak type estimates for singular integrals. Rev Mat Iberoam, 2017, 33(4):1267-1284
[4] Ding Y, Lai X. Weak type (1, 1) behavior for the maximal operator with L¹-Dini kernel. Potential Anal, 2017, 47(2):169-187
[5] Ding Y, Wu X. Littlewood-Paley g-functions with rough kernels on homogeneous groups. Studia Math, 2009, 195(1):51-86
[6] Fan D, Sato S. Weak type (1, 1) estimates for Marcinkiewicz integrals with rough kernels. Tohoku Math J, 2001, 53(2):265-284
[7] Garcia-Cuerva J, Rubio de Francia J. Weighted Norm Inequalities and Related Topics. Amsterdam:NorthHolland, 1985
[8] Guo W, He J, Wu H. Limiting weak-type behviors of some integral operators. arXiv:1710.10602v1
[9] Hörmander L. Estimates for translation invariant operators in L^p spaces. Acta Math, 1960, 104:93-139
[10] Hou X, Wu H. On the limiting weak-type behviors for maximal operators associated with power weighted measure. Canad Math Bull, doi:10.4153/CMB-2018-017-2(to appear)
[11] Janakiraman P. Limiting weak-type behavior for singular integral and maximal operators. Trans Amer Math Soc, 2006, 358(5):1937-1952
[12] Lu S, Ding Y, Yan D. Singular Integrals and Related Topics. Singapore:World Scientific Publishing Company, 2011
[13] Liu F. Integral operators of Marcinkiewicz type on Triebel-Lizorkin spaces. Math Nachr, 2017, 290(1):75-96
[14] Liu F. On the Triebel-Lizorkin space boundedness of Marcinkiewicz integrals along compound surfaces. Math Inequal Appl, 2017, 20(2):515-535
[15] Liu F, Wu H. L^p bounds for Marcinkiewicz integrals associated to homogeneous mappings. Monatsh Math, 2016, 181(4):875-906
[16] Liu F, Xue Q. Characterizations of multiple Littlewood-Paley operators on product domains. Publ Math Debrecen, 2018, 92(3/4):419-439
[17] Sato S. Estimates for Littlewood-Paley functions and extrapolation. Integral Equations Operator Theory, 2008, 63:429-440
[18] Stein E. On the functions of Littlewood-Paley, Lusin, and Marcinkiewicz. Trans Amer Math Soc, 1958, 88:430-466
[19] Wu H. L^p bounds for Marcinkiewicz integrals associated to surfaces of revolution. J Math Anal Appl, 2006, 321(2):811-827
[20] Wu H. On Marcinkiewicz integral operators with rough kernels. Integral Equations Operator Theory, 2005, 52(2):285-298
[21] Wu H. General Littlewood-Paley functions and singular integral operators on product spaces. Math Nachr, 2006, 279(4):431-444
[22] Wu H. A rough multiple Marcinkiewicz integral along continuous surfaces. Tohoku Math J, 2007, 59(2):145-166
[23] Wu H, Xu J. Rough Marcinkiewica integrals associated to surfaces of revolution on product domains. Acta Math Sci, 2009, 29B(2):294-304

LIMITING WEAK-TYPE BEHAVIORS FOR CERTAIN LITTLEWOOD-PALEY FUNCTIONS

LIMITING WEAK-TYPE BEHAVIORS FOR CERTAIN LITTLEWOOD-PALEY FUNCTIONS

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