Boundedness of some integral operators and commutators on homogeneous Herz spaces with three variable exponents

doi:10.1007/s11464-021-0897-6

Front. Math. China

2021, Vol. 16

Issue (1) : 211-237 https://doi.org/10.1007/s11464-021-0897-6

RESEARCH ARTICLE

Boundedness of some integral operators and commutators on homogeneous Herz spaces with three variable exponents

Xia YU, Zongguang LIU(

)

School of Sciences, China University of Mining and Technology, Beijing 100083, China

Download: PDF(335 KB)
Export: BibTeX | EndNote | Reference Manager | ProCite | RefWorks

Abstract

We obtain the boundedness of some integral operators and commutators on homogeneous Herz spaces with three variable exponents $K ˙ p (⋅), q (⋅) α (⋅)$ , such as some sublinear operators, the fractional integral and its commutator.

Keywords Sublinear operator fractional integral commutator homogeneous Herz space variable exponent

Corresponding Author(s): Zongguang LIU

Issue Date: 26 March 2021

Cite this article:

Xia YU,Zongguang LIU. Boundedness of some integral operators and commutators on homogeneous Herz spaces with three variable exponents[J]. Front. Math. China, 2021, 16(1): 211-237.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-021-0897-6
https://academic.hep.com.cn/fmc/EN/Y2021/V16/I1/211

1	A Almeida, D Drihem. Maximal, potential and singular type operators on Herz spaces with variable exponents. J Math Anal Appl, 2012, 394(2): 781–795 https://doi.org/10.1016/j.jmaa.2012.04.043
2	C Capone, D Cruz-Uribe, A Fiorenza. The fractional maximal operator and fractional integrals on variable Lp spaces. Rev Mat Iberoam, 2007, 23(23): 743–770 https://doi.org/10.4171/RMI/511
3	D Cruz-Uribe, A Fiorenza, J M Martell, C Pérez. The boundedness of classical operators on variable Lp spaces. Ann Acad Sci Fenn Math, 2006, 31(1): 239–264
4	D Cruz-Uribe, A Fiorenza, C Neugebauer. The maximal function on variable Lp spaces. Ann Acad Sci Fenn Math, 2003, 28(1): 223–238
5	L Diening, P Harjulehto, P Hästö, M Růžička. Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Math, Vol 2017. Berlin: Springer-Verlag, 2011 https://doi.org/10.1007/978-3-642-18363-8
6	C S Herz. Lipschitz spaces and Bernstein's theorem on absolutely convergent Fourier transforms. J Math Mech, 1968, 18: 283–323 https://doi.org/10.1512/iumj.1969.18.18024
7	M Izuki. Vector-valued inequalities on Herz spaces and characterizations of Herz-Sobolev spaces with variable exponent. Glas Mat, 2010, 45(2): 475–503 https://doi.org/10.3336/gm.45.2.14
8	M Izuki. Boundedness of commutators on Herz spaces with variable exponent. Rend Circ Mat Palermo, 2010, 59(2): 199–213 https://doi.org/10.1007/s12215-010-0015-1
9	M Izuki. Boundedness of sublinear operators on Herz spaces with variable exponent and application to wavelet characterization. Anal Math, 2010, 36(1): 33–50 https://doi.org/10.1007/s10476-010-0102-8
10	M Izuki. Commutators of fractional integrals on Lebesgue and Herz spaces with variable exponent. Rend Circ Mat Palermo, 2010, 59(3): 461–472 https://doi.org/10.1007/s12215-010-0034-y
11	M Izuki, T Noi. Boundedness of some integral operators and commutators on generalized Herz spaces with variable exponents. OCAMI Preprint Ser, 2011, 11–15
12	O Kováčik, J Rákosník. On spaces Lp(x) and Wk,p(x). Czechoslovak Math J, 1991, 41: 592–618 https://doi.org/10.21136/CMJ.1991.102493
13	X W Li, D C Yang. Boundedness of some sublinear operators on Herz spaces. Illinois J Math, 1996, 40(3): 484–501 https://doi.org/10.1215/ijm/1255986021
14	S Z Lu, D C Yang, G Hu. Herz Type Spaces and Their Applications.Beijing: Science Press, 2008
15	W Orlicz. On the summability of bounded sequences by continuous methods. Bull Acad Polon Sci Ser Sci Math Astr Phys, 1958, 6: 549{556
16	F Soria, G Weiss. A remark on singular integrals and power weights. Indiana Univ Math J, 1994, 43(1): 187–204 https://doi.org/10.1512/iumj.1994.43.43009

[1]	Shanzhen LU. Function characterizations via commutators of Hardy operator[J]. Front. Math. China, 2021, 16(1): 1-12.
[2]	Jingshi XU. Variable integral and smooth exponent Besov spaces associated to non-negative self-adjoint operators[J]. Front. Math. China, 2020, 15(6): 1245-1263.
[3]	Hongbin WANG, Jingshi XU, Jian TAN. Boundedness of multilinear singular integrals on central Morrey spaces with variable exponents[J]. Front. Math. China, 2020, 15(5): 1011-1034.
[4]	Rui BU, Zunwei FU, Yandan ZHANG. Weighted estimates for bilinear square functions with non-smooth kernels and commutators[J]. Front. Math. China, 2020, 15(1): 1-20.
[5]	Xing FU. Equivalent characterizations of Hardy spaces with variable exponent via wavelets[J]. Front. Math. China, 2019, 14(4): 737-759.
[6]	Sibei YANG, Dachun YANG, Wen YUAN. New characterizations of Musielak–Orlicz–Sobolev spaces via sharp ball averaging functions[J]. Front. Math. China, 2019, 14(1): 177-201.
[7]	Yanping CHEN, Liwei WANG. L²( $ℝ n$ ) boundedness for Calderón commutator with rough variable kernel[J]. Front. Math. China, 2018, 13(5): 1013-1031.
[8]	Huixia MO, Xiaojuan WANG, Ruiqing MA. Commutator of Riesz potential in p-adic generalized Morrey spaces[J]. Front. Math. China, 2018, 13(3): 633-645.
[9]	Nguyen Minh CHUONG, Nguyen Thi HONG, Ha Duy HUNG. Bounds of weighted multilinear Hardy-Cesàro operators in p-adic functional spaces[J]. Front. Math. China, 2018, 13(1): 1-24.
[10]	Wei WANG, Jingshi XU. Multilinear Calderón-Zygmund operators and their commutators with BMO functions in variable exponent Morrey spaces[J]. Front. Math. China, 2017, 12(5): 1235-1246.
[11]	Dongxiang CHEN, Shanzhen LU, Suzhen MAO. Multiple weighted estimates for maximal vector-valued commutator of multilinear Calderón-Zygmund singular integrals[J]. Front. Math. China, 2017, 12(3): 531-558.
[12]	Jing ZHANG,Huoxiong WU. Oscillation and variation inequalities for singular integrals and commutators on weighted Morrey spaces[J]. Front. Math. China, 2016, 11(2): 423-447.
[13]	Xiao YU,Shanzhen LU. Boundedness for a class of fractional integrals with a rough kernel related to block spaces[J]. Front. Math. China, 2016, 11(1): 173-187.
[14]	Wenjuan LI,Qingying XUE. Continuity properties for commutators of multilinear type operators on product of certain Hardy spaces[J]. Front. Math. China, 2014, 9(6): 1325-1347.
[15]	Chune SHI, Jingshi XU. Herz type Besov and Triebel-Lizorkin spaces with variable exponent[J]. Front Math Chin, 2013, 8(4): 907-921.

Viewed

Full text

Abstract

Cited

Shared

Discussed