JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2016, Vol. 51 ›› Issue (4): 9-18.doi: 10.6040/j.issn.1671-9352.0.2015.123

Previous Articles     Next Articles

Boundedness of the Littlewood-Paley operators and cummutators on the Herz spaces with variable exponents

WANG Jie, QU Meng, SHU Li-sheng   

  1. School of Mathematics and Computer Science, Anhui Normal University, Wuhu 241003, Anhui, China
  • Received:2015-03-24 Online:2016-04-20 Published:2016-04-08

PDF (PC)

1067

Abstract: The main result in the paper is the boundedness of the Littlewood-Paley integral operators(include Lusin area integral function, Littlewood-Paley g and g*λ)and its higher order commutators generated by BMO fucntions on the Herz spaces with two variable exponents p(·),α(·), where p(·),α(·) satisfies some continuous condition.

Key words: variable exponent, Littlewood-Paley integral operator, higher order commutator, Herz space

CLC Number: 

  • O174.2
[1] LU Shanzhen, YANG Dachun. The central BMO spaces and Littlewood-Paley operators[J]. Approx Theory Appl, 1995, 11(3):72-94.
[2] LI Xinwei, YANG Dachun. Boundedness of some sublinear operators on Herz spaces[J]. Illinois Journal of Mathematics, 1996, 40(3):484-501.
[3] LIU Lanzhe, LU Shanzhen, XU Jingshi. Boundedness of commutators of Littlewood-Paley operators[J]. Andvances in Mathematics, 2003, 32(4):473-480.
[4] 陈杰诚,丁勇,范大山.带变量核的Littlewood-Paley算子[J]. 中国科学:A辑数学, 2006, 36(1):38-51. CHEN Jiecheng, DING Yong, FAN Dashan.Littlewood-Paley operator with variable kernel[J].Science in China: Series A Mathematics, 2006, 36(1):38-51.
[5] 陈艳萍,丁勇.广义Morrey空间上带变量核的的Littlewood-Paley算子[J]. 数学物理学报, 2009, 29A(3):630-642. CHEN Yanping, DING Yong. Littlewood-Paley operator with variable kernel on generalized Morrey spaces[J]. Acta Mathematica: Scientia-Chinese series, 2009, 29A(3):630-642.
[6] XUE Qingying, DING Yong. Weighted estimates for multilinear commutators of the Littlewood-Paley operators[J].Science in China: Series A Mathematics, 2009, 52(9):1849-1868.
[7] Kovácik O, Rákosník J. On spaces Lp(x) and Wk,p(x)[J]. Czechoslovak Math J, 1991, 41(4):592-618.
[8] CRUZ-URIBE SFO D, FIORENZA A, MARTELL M J, et al. The boundedness of classical operators on variable Lp spaces[J]. Ann Acad Sci Fenn Math, 2006, 31(1):239-264.
[9] WANG Hongbin, FU Zunwei, LIU Zongguang. Higher-order commutators of Marcinkiewicz integrals on variable Lebesgue spaces[J]. Acta Math Sci, 2012, 32A(6):1092-1101.
[10] IZUKI M. Herz and amalgam spaces with variable exponent, the Haar wavelets and greediness of the wavelet system[J]. East Journal on Approximations, 2009, 15(1):87-109.
[11] IZUKI M. Commutators of fractional integrals on Lebesgue and Herz spaces with variable exponent[J]. Rend Circ Mat Palermo, 2010, 59(3):461-472.
[12] ALMEIDA A, DRIHEM D. Maximal, potential and singular type operators on Herz spaces with variable exponents[J]. J Math Anal Appl, 2012, 394(2):781-795.
[13] LU Yan, ZHU Yueping. Boundedness of multilinear Calderón-Zygmund singular operators on Morrey-Herz spaces with variable exponents[J]. Acta Mathematica Sinica: English Series, 2014, 30(7):1180-1194.
[14] WANG Lijuan, TAO Shuangping. Boundedness of Littlewood-Paley operators and their commutators on Herz-Morrey spaces with exponent[J]. Journal of Inequalities and Applications, 2014, 2014(1):227.
[15] DONG Baohua, XU Jingshi. New Herz type Besov and Triebel-Lizorkin spaces with variable exponents [J]. J of Function Spaces and Applications, 2012, 27. pages, id: 384593.
[16] IZUKI M. Vector-valued inequalities on Herz spaces and characterizations of Herz-Sobolev spaces with variable exponent[J]. Glasnik Mat, 2010, 45(2):475-503.
[17] WANG Liwei. Marcinkiewicz integral operators and commutators on Herz spaces with variable exponents[J]. J Function Space and Applications, 2014, id:430635.
[1] XIN Yin-ping, TAO Shuang-ping. Boundedness of Marcinkiewicz integrals operators with variable kernels on Herz-type Hardy spaces with variable exponent [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(6): 38-43.
[2] ZHANG Shen-gui. Multiple solutions of Navier boundary value problem for fourth-order elliptic equation with variable exponents [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(2): 32-37.
[3] ZHAO Huan, ZHOU Jiang. Commutators of multilinear Calderón-Zygmund operator on Herz-type Hardy spaces with variable exponent [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(10): 42-50.
[4] LU Qiang-de, TAO Shuang-ping. Boundedness of commutators of Calderón-Zygmund operators and fractional integrals in homogeneous grand variable exponent Lebesgue spaces [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(9): 54-58.
[5] MA Xiao-jie, ZHAO Kai. Boundedness of commutators of the fractional Hardy operators on weighted spaces with variable exponent [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(11): 106-110.
[6] WANG Jin-ping, ZHAO Kai. Lipschitz commutators of fractional integrals on Herz-type spaces with variable exponents [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2016, 51(10): 6-10.
[7] ZHAO Kai, JI Chun-jing, HUANG Zhi. Boundedness of the commutator of some Marcinkiewicz integral on Herz-type Hardy spaces [J]. J4, 2013, 48(6): 1-4.
[8] ZHAO Kai, SUN Xiao-hua, DU Hong-bin. Weak boundedness of Littlewood-Paley operators on  weighted Herz spaces [J]. J4, 2012, 47(4): 62-65.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
No Suggested Reading articles found!
Copyright 2018 © Editorial office of JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)
Supported by Beijing Magtech Co.Ltd