Some variable exponent boundedness and commutators estimates for fractional Rough Hardy operators on central Morrey space

Abstract

In this article, we study the boundedness of the fractional Rough Hardy operator and its adjoint operators on the central Morrey space with a variable exponent. We also establish the same boundedness for their commutators when the symbol functions are on the λ-central BMO space with a variable exponent.

Keywords

• Rough Hardy-type operators
• central Morrey space
• fractional integral
• variable exponent

References

1. Hardy, G. H., Note on a theorem of Hilbert, Math. Z., 6 (1920), 314-317.https://doi.org/10.1007/BF01199965
2. Faris, W. G., Weak Lebesgue spaces and quantum mechanical binding, Duke Math. J., 43(4) (1976), 365-373. https://doi.org/10.1215/S0012-7094-76-04332-5
3. Christ, M., Grafakos, L., Best constants for two non convolution inequalities, Proc. Amer. Math. Soc., 123 (1995), 1687-1693. https://doi.org/10.2307/2160978
4. Sawyer, E., Weighted Lebesgue and Lorentz norm inequalities for the Hardy operator, Trans. Amer. Math. Soc., 281 (1984), 329-337. https://doi.org/10.2307/1999537
5. Fu, Z., Liu, Z., Lu, S., Wang, H., Charactrization for commutators of n-dimensional fractional Hardy operators, Sci. China Ser. A., 50 (2007), 1418-1426.https://doi.org/10.1007/s11425-007-0094-4.
6. Ren, Z., Tao, S., Weighted estimates for commutators of n-dimensional rough hardy operators, J. Funt. Spaces., (2013), 1-13. https://doi.org/10.1155/2013/568202
7. Fu, Z., Lu, S., Zhao, F., Commutators of n-dimensional rough Hardy operators, Sci. China Ser. A., 54(2011), 95-104. https://doi.org/10.1007/s11425-010-4110-8
8. Orlicz, W., Über konjugierte exponentenfolgen, Studia Math., 3(1931), 200-212. https://doi.org/10.4064/SM-3-1-200-211

9. Nakano, H., Modulared Semi-Ordered Linear Spaces, Maruzen Co, Ltd, Tokyo, 1951.
10. Uribe, D. C., Fiorenza, A., Martell, J. M., Pérez, C., The boundedness of classical operators on variable Lp spaces, Ann. Acad. Sci. Fenn. Math., 31(2006), 239-264.
11. Dining, L., Reisz potential and Soblev embedding on genealized Lesbesgue and Sobolev $L^{p(·)}$ and $W^{k,p(·)}$, Math. Nachr., 268 (2004), 31-43. https://doi.org/10.1002/mana.200310157
12. Uribe, D. C., Fiorenza, A., Neugebauer, A., The maximal function on variable $L^{p}$ spaces, Ann. Acad. Sci. Fenn. Math., 28 (2003), 223-238.
13. Uribe, D. C., Diening, L., Fiorenza, A., A new proof of the boundedness of maximal operators on variable Lebesgue spaces, Boll. Unione Mat. Ital., 2 (2009), 151-173. http://eudml.org/doc/290576
14. Alvarez, J., Lakey, J., Partida, M. G., Spaces of bounded λ-central mean oscillation, Morrey spaces, and λ-central Carleson measures, Collect. Math., 51(2000), 1-47.
15. Morrey, C., On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc., 43(1938), 126-166. https://doi.org/10.1090/S0002-9947-1938-1501936-8
16. Chuong, N., Duong, D., Hung, H., Bounds for the weighted Hardy-Cesaro operator and its commutator on Morrey-Herz type spaces, Z. Anal. Anwend., 35 (2016), 489-504. https://doi.org/10.4171/ZAA/1575
17. Wu, Q., Fu, Z., Boundedness of Hausdorff operators on Hardy spaces in the Heisenberg group, Banach J. Math. Anal., 12 (2018), 909-934. https://doi.org/10.1215/17358787-2018-0006
18. Chen, Y., Levin, S., Rao, M., Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66 (2006), 1383-1406. https://doi.org/10.1137/050624522
19. Ruicka, M., Electrorheogical Fluid: Modeling and Mathematical Theory, Springer, Berlin, 2000.
20. Yang, M., Fu, Z., Sun, J., Global solutions to Chemotaxis-Navier-Stokes equations in critical Besov spaces, Dis. Contin. Dyn. Syst. Ser. B., 23 (2018), 3427-3460. https://doi.org/10.3934/dcdsb.2018284
21. Kováčik, O., Rákosník, J., On spaces $L^{p(x)}$ and $W^{k,p(x)}$, Czechoslovak Math. J., 41 (1991), 592-618. https://doi.org/10.21136/CMJ.1991.102493
22. Mizuta, Y., Ohno, T., Shimomura, T., Boundedness of maximal operators and Sobolev’s theorem for non-homogeneous central Morrey spaces of variable exponent, Hokkaido Math. J., 44 (2015), 185-201. https://doi.org/10.14492/hokmj/1470053290
23. Wang, D., Liu, Z., Zhou, J., Teng, Z., Central BMO spaces with variable exponent, arXiv:1708.00285, 2017.
24. Fu, Z., Lu, S., Wang, H., Wang, L., Singular integral operators with rough kernel on central Morrey spaces with variable exponent, Ann. Acad. Sci. Fenn. Math., 44 (2019), 505-522. https://doi.org/10.5186/aasfm.2019.4431
25. Hussain, A., Asim, M., Commutators of the fractional Hardy operator on weighted variable Herz-Morrey spaces, J. Funt. Space.., ID 9705250(2021), 10 pages. doi.org/10.1155/2021/9705250.
26. Hussain, A., Asim, M., Variable λ-central Morrey space estimates for the fractional Hardy operators and commutators, J. Math., ID 5855068(2022), 12 pages. https://doi.org/10.1155/2022/5855068
27. Asim, M., Hussain, A., Weighted variable Morrey-Herz estimates for fractional Hardy operators, J. Inq. Appl., 2(2022) (2022) 12pp. doi.org/10.1186/s13660-021-02739-z
28. Huang, A., Xu, J., Multilinear singular integrals and commutators in variable exponent Lebesgue spaces, Appl. Math. J. Chin. Univ., 25 (2010), 69-77. https://doi.org/10.1007/s11766-010-2167-3
29. Asim, M., Ayoob, I., Weighted estimates for fractional bilinear Hardy operators on variable exponent Morrey-Herz space, J. Inq. Appl., 11(2024) 2024 19pp. doi.org/10.1186/s13660-024-03092-7
30. Jianglong, W., Boundedness of some sublinear operators on Herz-Morrey spaces with variable exponent, Georgian Math. J., 21 (2014), 101-111. https://doi.org/10.1515/gmj-2014-0004
31. Wu, J. L., Zhao, W. J., Boundedness for fractional Hardy-type operator on variableexponent Herz-Morrey spaces, Kyoto J. Math., 56 (2016), 831-845. https://doi.org/10.1215/21562261-3664932
32. Nekavinda, A., Hardy-Littlewood maximal operator on $L^{p(x)}(R)$, Math. Inequal. Appl., 7 (2004), 255-265. https://doi.org/10.7153/mia-07-28
33. Diening, L., Maximal functions on Musielak-Orlicz spaces and generalized Lebesgue spaces, Bull. Sci. Math., 129 (2005), 657-700. https://doi.org/10.1016/j.bulsci.2003.10.003
34. Izuki, M., Fractional integrals on Herz-Morrey spaces with variable exponent, Hiroshima Math. J., 40 (2010), 343-355. https://doi.org/10.32917/hmj/1291818849.
35. Grafakos, L., Modern Fourier Analysis , 2nd edition, Springer, 2009.
36. Izuki, M., Boundedness of commutators on Herz spaces with variable exponent, Rendiconti del Circolo Matematico di Palermo., 59 (2010), 199-213. https://doi.org/10.1007/s12215-010-0015-1
37. Capone, C., Uribe, D. C., Fiorenza, A., The fractional maximal operator and fractional integrals on variable Lp(R) spaces, Rev. Mat. Iberoam., 23 (2007), 743-770. https://doi.org/10.4171/RMI/511