Boundedness for variable fractional integral operators and their commutators on Herz–Hardy spaces with variable exponent

Let $Esubset mathbb{R}^{n}$ be a bounded open set. In this paper, we establish the boundedness of variable fractional integral operators and their commutators on variable Herz–Hardy spaces $Hdot{K}_{p(.)}^{alpha (.),q(.)}$ with three variable exponents by using the atomic decomposition.

PDF

___

[1] Almeida A, Drihem D. Maximal, potential and singular type operators on Herz spaces with variable exponents. Journal of Mathematical Analysis and Applications 2012; 394 (2): 781-795. doi: 110.1016/j.jmaa.2012.04.043
[2] Almeida A, Hasanov J, Samko S. Maximal and potential operators in variable exponent Morrey spaces. Georgian Mathematical Journal 2008; 15 (2): 195-208. doi: 10.1515/GMJ.2008.195
[3] Bilalov BT, Guseynov ZG. Basicity of a system of exponents with a piece-wise linear phase in variable spaces. Mediterranean Journal of Mathematics 2012; 9 (3): 487-498. doi: 10.1007/s00009-011-0135-7
[4] Cheng XX, Shu LS. Boundedness for some Hardy type operators on Herz-Morrey spaces with variable exponent. Journal of Anhui Normal University (Natural Sciences) 2015; 38 (1): 19-24. doi: 10.14182/J.cnki.1001- 2443.2015.01.005
[5] Cruz-Uribe D, Fiorenza A. Variable Lebesgue Spaces. Heidelberg, Germany: Birkhäuser/Springer, 2013. doi: 10.1007/978-3-0348-0548-3
[6] Diening L, Harjulehto P, Hästö P, Ruzicka M. Lebesgue and Sobolev Spaces with Variable Exponents. Heidelberg, Germany: Springer, 2011. doi: 10.1007/978-3-642-18363-8
[7] Diening L, Ruzicka M. Calderón-Zygmund operators on generalized Lebesgue spaces Lp(·) and problems related to fluid dynamics. Journal für die Reine und Angewandte Mathematik 2003; 563: 197-220. doi: 10.1515/crll.2003.081
[8] Drihem D, Seghiri F. Notes on the Herz-type Hardy spaces of variable smoothness and integrability. Mathematical Inequalities Applications 2016; 19 (1): 145-165. doi: 10.7153/mia-19-11
[9] Heraiz R. Variable Herz estimates for fractional integral operators. Ukrainian Mathematical Journal 2021; 72 (8): 1197-1211. doi: 10.1007/s11253-020-01857-z
[10] Herz CS. Lipschitz spaces and Bernstein’s theorem on absolutely convergent Fourier transforms. Journal of Mathmatics and Mechanics 1968; 18: 283-323.
[11] Izuki M. Boundedness of sublinear operators on Herz spaces with variable exponent and application to wavelet characterization. Analysis Mathematica 2010; 36 (1): 33-50. doi: 10.1007/s10476-010-0102-8
[12] Izuki M. Boundedness of commutators on Herz spaces with variable exponent. Rendiconti del Circolo Matematico di Palermo. Second Series 2010; 59 (2): 199-213. doi: 10.1007/s12215-010-0015-1
[13] Izuki M, Noi T. Boundedness of some integral operators and commutators on generalized Herz spaces with variable exponents. Preprint (2011).
[14] Kováčik O, Rákosnik J. On spaces $L ^{p(x)} and W^{k,p(x)}$. Czechoslovak Mathematical Journal 1991; 41 (4): 592-618. doi: 10.21136/CMJ.1991.102493
[15] Lu SZ, Yang DC, Hu GE. Herz Type Spaces and Their Applications. Beijing, China: Science Press, 2008. 1151 XIN/Turk J Math
[16] Nakai E, Sawano Y. Hardy spaces with variable exponents and generalized Campanato spaces. Journal of Functional Analysis 2012; 262 (9): 3665-3748. doi: 10.1016/j.jfa.2012.01.004
[17] Sawano Y, Ho KP, Yang DC, Yang SB. Hardy spaces for ball quasi-Banach function spaces. Dissertationes Mathematicae 2017; 525: 1-102. doi: 10.4064/dm750-9-2016
[18] Sharapudinov II. On direct and inverse theorems of approximation theory in variable Lebesgue and Sobolev spaces. Azerbaijan Journal of Mathematics 2014; 4 (1): 55-72.
[19] Tang CQ, Wu Q, Xu JS. Estimates of fractional integral operators on variable exponent Lebesgue spaces. Journal of Function Spaces 2016; 2016: 1-7. doi: 10.1155/2016/2438157
[20] Wang HB, Liu ZG. The Herz-type Hardy spaces with variable exponent and their applications. Taiwanese Journal of Mathematics 2012; 16 (4): 1363-1389. doi: 10.11650/twjm/1500406739
[21] Welland GV. Weighted norm inequalities for fractional integrals. Proceedings of the American Mathematical Society 1975; 51 (1): 143-148. doi: 10.1090/S0002-9939-1975-0369641-X
[22] Xin YP, Tao SP. Boundedness of Marcinkiewicz integrals operators with variable kernels on Herz-type Hardy spaces with variable exponent. Journal of Shandong University (Natural Sciences) 2018; 53 (6): 38-43. doi: 10.6040/j.issn.1671-9352.0.2017.646