BOUNDEDNESS FOR FRACTIONAL HARDY-TYPE OPERATOR ON HERZ-MORREY SPACES WITH VARIABLE EXPONENT

Title & Authors
BOUNDEDNESS FOR FRACTIONAL HARDY-TYPE OPERATOR ON HERZ-MORREY SPACES WITH VARIABLE EXPONENT
Wu, Jianglong;

Abstract
In this paper, the fractional Hardy-type operator of variable order $\small{{\beta}(x)}$ is shown to be bounded from the Herz-Morrey spaces $\small{M\dot{K}^{{\alpha},{\lambda}}_{p_1,q_1({\cdot})}(\mathbb{R}^n)}$ with variable exponent $\small{q_1(x)}$ into the weighted space $\small{M\dot{K}^{{\alpha},{\lambda}}_{p_2,q_2({\cdot})}(\mathbb{R}^n,{\omega})}$, where ${\omega} Keywords Herz-Morrey space;Hardy operator;Riesz potential;variable exponent;weighted estimate; Language English Cited by 1. Boundedness for Higher Order Commutators of Fractional Integrals on Variable Exponent Herz–Morrey Spaces, Mediterranean Journal of Mathematics, 2017, 14, 5 References 1. A. Almeida and D. Drihem, Maximal, potential and singular type operators on Herz spaces with variable exponents, J. Math. Anal. Appl. 39 (2012), no. 2, 781-795. 2. M. Christ and L. Grafakos, Best constants for two nonconvolution inequalities, Proc. Amer. Math. Soc. 123 (1995), no. 6, 1687-1693. 3. Z. Fu, Z. Liu, S. Lu, and H. Wang, Characterization for commutators of n-dimensional fractional Hardy operators, Sci. China Ser. A 50 (2007), no. 10, 1418-1426. 4. J. L. Wu, Boundedness of multilinear commutators of fractional Hardy operators, Acta Math. Sci. Ser. A Chin. Ed. 31 (2011), no. 4, 1055-1062. 5. J. L. Wu and Q. G. Liu,${\lambda}$-central BMO estimates for higher order commutators of Hardy operators, Commun. Math. Res. In Prss. 6. J. L. Wu and J. M. Wang, Boundedness of multilinear commutators of fractional Hardy operators, Appl. Math. J. Chinese Univ. Ser. A 25 (2010), no. 1, 115-121. 7. J. L. Wu and P. Zhang, Boundedness of commutators of the fractional Hardy operators on Herz-Morrey spaces with variable exponent, Adv. Math. (China), In Press. 8. J. L. Wu and P. Zhang, Boundedness of multilinear Hardy type operators on product of Herz-Morrey spaces with variable exponent, Appl. Math. J. Chinese Univ. Ser. A 28 (2013), no. 2, 154-164. 9. P. Zhang and J. L. Wu, Boundedness of fractional Hardy type operators on Herz-Morrey spaces with variable exponent, J. Math. Practice Theory 42 (2013), no. 7, 247-254. 10. W. Orlicz, Uber konjugierte exponentenfolgen, Studia Math. 3 (1931), no. 3, 200-212. 11. H. Nakano, Modulared Semi-Ordered Linear Spaces, Maruzen Co. Ltd., Tokyo, 1950. 12. H. Nakano, Topology of Linear Topological Spaces, Maruzen Co. Ltd., Tokyo, 1951. 13. M. Ruzicka, Electrorheological Fluids: Modeling and Mathematical Theory, New York, Springer, vol. 1748, Lecture Notes in Math., 2000. 14. L. Diening and M. Ruzicka, Calderon-Zygmund operators on generalized Lebesgue spaces$L^{p({\cdot})}$and problems related to fluid dynamics, J. Reine Angew. Math. 563 (2003), 197-220. 15. O. Kovacik and J. Rakosnik, On spaces$L^{p(x)}$and$W^{k,p(x)}$, Czechoslovak Math. J. 41 (1991), no. 4, 592-618. 16. D. Cruz-Uribe, L. Diening, and A. Fiorenza, A new proof of the boundedness of maximal operators on variable Lebesgue spaces, Boll. Unione Mat. Ital. 2 (2009), no. 1, 151-173. 17. D. Cruz-Uribe, A. Fiorenza, J. Martell, and C. Perez, The boundedness of classical operators on variable$L^p$spaces, Ann. Acad. Sci. Fenn. Math. 31 (2006), no. 1, 239-264. 18. D. Cruz-Uribe, A. Fiorenza, and C. Neugebauer, The maximal function on variable$L^p$spaces, Ann. Acad. Sci. Fenn. Math. 28 (2003), no. 1, 223-238. 19. L. Diening, Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spaces$L^{p({\cdot})}$and$W^{k,p({\cdot})}$, Math. Nachr. 268 (2004), 31-43. 20. L. Diening, Maximal functions on generalized Lebesgue spaces$L^{p({\cdot})}$, Math. Inequal. Appl. 7 (2004), no. 2, 245-253. 21. L. Diening, Maximal functions on Musielak-Orlicz spaces and generalized Lebesgue spaces, Bull. Sci. Math. 129 (2005), no. 8, 657-700. 22. L. Diening, P. Harjulehto, P. Hasto, Y. Mizuta, and T. Shimomura, Maximal functions in variable exponent spaces: limiting cases of the exponent, Ann. Acad. Sci. Fenn. Math. 34 (2009), no. 2, 503-522. 23. M. Izuki, Fractional integrals on Herz-Morrey spaces with variable exponent, Hiroshima Math. J. 40 (2010), no. 3, 343-355. 24. V. Kokilashvili and S. Samko, On Sobolev theorem for Riesz type potentials in Lebesgue spaces with variable exponent, Z. Anal. Anwendungen 22 (2003), no. 4, 899-910. 25. T. Kopaliani, Infimal convolution and Muckenhoupt$A_{p({\cdot})}$condition in variable$L^p$spaces, Arch. Math. 89 (2007), no. 2, 185-192. 26. A. Lerner, On some questions related to the maximal operator on variable$L^p$spaces, Trans. Amer. Math. Soc. 362 (2010), no. 8, 4229-4242. 27. A. Nekvinda, Hardy-Littlewood maximal operator on$L^{p(x)}(\mathbb{R}^n)$, Math. Inequal. Appl. 7 (2004), no. 2, 255-265. 28. L. Pick and M. Ruzicka, An example of a space$L^{p({\cdot})}$on which the Hardy-Littlewood maximal operator is not bounded, Expo. Math. 19 (2001), no. 4, 369-371. 29. S. Samko, Convolution and potential type operators in$L^{p(x)}(\mathbb{R}^n)\$, Integral Transform. Spec. Funct. 7 (1998), no. 3-4, 261-284.