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

Abstract

We obtain the boundedness of some integral operators and commutators on homogeneous Herz spaces with three variable exponents {{\displaystyle \dot{K}}}_{p(\cdot ),q(\cdot )}^{\alpha (\cdot )}, such as some sublinear operators, the fractional integral and its commutator.

Key words： Sublinear operator; fractional integral; commutator; homogeneous Herz space; variable exponent

References

1 Almeida A, Drihem D. Maximal, potential and singular type operators on Herz spaces with variable exponents. J Math Anal Appl, 2012, 394(2): 781–795

2 Capone C, Cruz-Uribe D, Fiorenza A. The fractional maximal operator and fractional integrals on variable L^p spaces. Rev Mat Iberoam, 2007, 23(23): 743–770

3 Cruz-Uribe D, Fiorenza A, Martell J M, Pérez C. The boundedness of classical operators on variable L^p spaces. Ann Acad Sci Fenn Math, 2006, 31(1): 239–264

4 Cruz-Uribe D, Fiorenza A, Neugebauer C. The maximal function on variable L^p spaces. Ann Acad Sci Fenn Math, 2003, 28(1): 223–238

5 Diening L, Harjulehto P, Hästö P, Růžička M. Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Math, Vol 2017. Berlin: Springer-Verlag, 2011

6 Herz C S. Lipschitz spaces and Bernstein's theorem on absolutely convergent Fourier transforms. J Math Mech, 1968, 18: 283–323

7 Izuki M. Vector-valued inequalities on Herz spaces and characterizations of Herz-Sobolev spaces with variable exponent. Glas Mat, 2010, 45(2): 475–503

8 Izuki M. Boundedness of commutators on Herz spaces with variable exponent. Rend Circ Mat Palermo, 2010, 59(2): 199–213

9 Izuki M. Boundedness of sublinear operators on Herz spaces with variable exponent and application to wavelet characterization. Anal Math, 2010, 36(1): 33–50

10 Izuki M. Commutators of fractional integrals on Lebesgue and Herz spaces with variable exponent. Rend Circ Mat Palermo, 2010, 59(3): 461–472

11 Izuki M, Noi T. Boundedness of some integral operators and commutators on generalized Herz spaces with variable exponents. OCAMI Preprint Ser, 2011, 11–15

12 Kováčik O, Rákosník J. On spaces L^p(x) and W^k,p(x). Czechoslovak Math J, 1991, 41: 592–618

13 Li X W, Yang D C. Boundedness of some sublinear operators on Herz spaces. Illinois J Math, 1996, 40(3): 484–501

14 Lu S Z, Yang D C, Hu G. Herz Type Spaces and Their Applications.Beijing: Science Press, 2008

15 Orlicz W. On the summability of bounded sequences by continuous methods. Bull Acad Polon Sci Ser Sci Math Astr Phys, 1958, 6: 549{556

16 Soria F, Weiss G. A remark on singular integrals and power weights. Indiana Univ Math J, 1994, 43(1): 187–204