本征平方函数在变指数Herz及Herz-Hardy空间上的有界性

数学物理学报 ›› 2018, Vol. 38 ›› Issue (4): 716-727.

本征平方函数在变指数Herz及Herz-Hardy空间上的有界性

王立伟¹, 束立生²

1 安徽工程大学数理学院安徽芜湖 241000;
2 安徽师范大学数学计算机科学学院安徽芜湖 241003

收稿日期:2017-08-09 修回日期:2017-12-18 出版日期:2018-08-26 发布日期:2018-08-26
通讯作者: 王立伟,E-mail:wangliwei@ahpu.edu.cn E-mail:wangliwei@ahpu.edu.cn
作者简介:束立生,E-mail:shulsh@mail.ahnu.edu.cn
基金资助:
国家自然科学基金（11771223）

Boundedness of the Intrinsic Square Function on Variable Exponent Herz and Herz-Hardy Spaces

Wang Liwei¹, Shu Lisheng²

1 School of Mathematics and Physics, Anhui Polytechnic University, Anhui Wuhu 241000;
2 School of Mathematics and Computer Science, Anhui Normal University, Anhui Wuhu 241003

Received:2017-08-09 Revised:2017-12-18 Online:2018-08-26 Published:2018-08-26
Supported by:
Supported by the NSFC (11771223)

摘要/Abstract

摘要： 该文证明了本征平方函数S_β在变指数Herz空间K_p（·）^{α（·），q}（Rⁿ）及Herz-Hardy空间HK_p（·）^{α（·），q}（Rⁿ）上的有界性，其中α（·）和p（·）均为变指数.特别地，当α（·）≡ α为常数时，相应结果也是新的.

关键词: 本征平方函数, 变指数Herz空间, 变指数Herz-Hardy空间

Abstract: In this paper, we show that the intrinsic square function S_β is bounded on the variable exponent Herz spaces K_p(·)^{α(·), q}(Rⁿ) and Herz-Hardy spaces HK_p(·)^{α(·), q}(Rⁿ), where α(·) and p(·) are variable. In particular, when α(·) ≡ α is constant, the corresponding main results are also new.

Key words: The intrinsic square function, Variable Herz spaces, Variable Herz-Hardy spaces

中图分类号:

O174.2

王立伟, 束立生. 本征平方函数在变指数Herz及Herz-Hardy空间上的有界性[J]. 数学物理学报, 2018, 38(4): 716-727.

Wang Liwei, Shu Lisheng. Boundedness of the Intrinsic Square Function on Variable Exponent Herz and Herz-Hardy Spaces[J]. Acta mathematica scientia,Series A, 2018, 38(4): 716-727.

参考文献 30

[1]	Wilson M. The intrinsic square function. Rev Mat Iberoam, 2007, 23:771-791
[2]	Fefferman C, Stein E M. Some maximal inequalities. Amer J Math, 1971, 93:107-115
[3]	Liang Y, Yang D. Intrinsic square function characterizations of Musielak-Orlicz Hardy spaces. Trans Amer Math Soc, 2015, 367:3225-3256
[4]	Zhuo C, Yang D, Liang Y. Intrinsic square function characterizations of Hardy spaces with variable exponents. Bull Malays Math Sci Soc, 2016, 39:1541-1577
[5]	Ho K P. Intrinsic square functions on Morrey and Block spaces with variable exponents. Bull Malays Math Sci Soc, 2017, 40:995-1010
[6]	Wilson M. Weighted Littlewood-Paley Theory and Exponential-Square Integrability. Berlin:SpringerVerlag, 2007
[7]	Kováčik O, Rákosník J. On spaces Lp(x) and Wk,p(x). Czechoslovak Math J, 1991, 41:592-618
[8]	Cruz-Uribe D, Fiorenza A, Neugebauer C. The maximal function on variable Lp spaces. Ann Acad Sci Fenn Math, 2003, 28:223-238
[9]	Diening L. Maximal function on generalized Lebesgue spaces Lp(·). Math Inequal Appl, 2004, 7:245-253
[10]	Harjulehto P, Hästö P, Lê Ú V, Nuortio M. Overview of differential equations with non-standard growth. Nonlinear Anal, 2010, 72:4551-4574
[11]	Nekvinda A. Hardy-littlewood maximal operator in Lp(x)(Rn). Math Inequal Appl, 2004, 7:255-265
[12]	Tan J, Liu Z, Zhao J. On some multilinear commutators in variable Lebesgue spaces. J Math Inequal, 2017, 11:715-734
[13]	Zhang P, Wu J. Commutators for the maximal function on Lebesgue spaces with variable exponent. Math Inequal Appl, 2014, 17:1375-1386
[14]	Almeida A, Drihem D. Maximal, potential and singular type operators on Herz spaces with variable exponents. J Math Anal Appl, 2012, 394:781-795
[15]	Dong B, Xu J. Variable Herz-type Hardy spaces and their applications. Anal Theory Appl, 2015, 31:321-353
[16]	Dong B, Xu J. New Herz type Besov and Triebel-Lizorkin spaces with variable exponents. J Funct Space Appl, 2012, Article ID:384593
[17]	Izuki M. Commutators of fractional integrals on Lebesgue and Herz spaces with variable exponent. Rend Circ Mat Palermo, 2010, 59:461-472
[18]	Lu Y, Zhu Y. Boundedness of multilinear Calderón-Zygmund singular operators on Morrey-Herz spaces with variable exponents. Acta Math Sin (Engl Ser), 2014, 30:1180-1194
[19]	Tao S, Wang L. Littlewood-Paley operators on Morrey spaces with variable exponent. The Scientific World Journal, 2014, Article ID:790671
[20]	Wang H, Liu Z. The Herz-type Hardy spaces with variable exponent and thier applications. Taiwanese J Math, 2012, 16:1363-1389
[21]	Yang D, Zhuo C, Yuan W. Besov-type spaces with variable smoothness and integrability. J Funct Anal, 2015, 269:1840-1898
[22]	Cruz-uribe D, Fiorenza A. Variable Lebesgue Spaces:Foundations and Harmonic Analysis, Applied and Numerical Harmonic Analysis. Boston:Birkhäuser Basel, 2013
[23]	Diening L, Harjulehto P, Hästö P, R·užička M. Lebesgue and Sobolev Spaces with Variable Exponents. Heidelberg:Springer, 2011
[24]	Wang H. The boundedness of intrinsic square functions on the weighted Herz spaces. J Funct Spaces, 2014, Article ID:274521
[25]	Wang H. Some estimates of intrinsic square functions on the weighted Herz-type Hardy spaces. J Inequal Appl, 2015, 2015(1):62
[26]	Diening L. Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces. Bull Sci Math, 2005, 129:657-700
[27]	Fefferman C, Stein E M. Hp spaces of several variables. Acta Math, 1972, 129:137-193
[28]	Lu S, Yang D, Hu G. Herz Type Spaces and Their Applications. Beijing:Science Press, 2008
[29]	Nakai E, Sawano Y. Hardy spaces with variable exponents and generalized Campanato spaces. J Funct Anal, 2012, 262:3665-3748
[30]	Yan X, Yang D, Yuan W, Zhuo C. Variable weak Hardy spaces and their applications. J Funct Anal, 2016, 271:2822-2887