In this paper, we obtain some boundedness of the $ n $-dimensional fractional Hardy operators with rough kernels and their commutators on central Morrey spaces with variable exponents.
Citation: Chenchen Niu, Hongbin Wang. $ N $-dimensional fractional Hardy operators with rough kernels on central Morrey spaces with variable exponents[J]. AIMS Mathematics, 2023, 8(5): 10379-10394. doi: 10.3934/math.2023525
In this paper, we obtain some boundedness of the $ n $-dimensional fractional Hardy operators with rough kernels and their commutators on central Morrey spaces with variable exponents.
[1] | A. Abdalmonem, A. Scapellato, Intrinsic square functions and commutators on Morrey-Herz spaces with variable exponents, Math. Meth. Appl. Sci., 44 (2021), 12408–12425. http://doi.org/10.1002/mma.7487 doi: 10.1002/mma.7487 |
[2] | A. O. Akdemir, M. T. Ersoy, H. Furkan, M. A. Ragusa, Some functional sections in topological sequence spaces, J. Funct. Spaces, 2022 (2022), 6449630. http://doi.org/10.1155/2022/6449630 doi: 10.1155/2022/6449630 |
[3] | J. Alvarez, J. Lakey, M. Guzmán-Partida, Spaces of bounded $\lambda$-central mean oscillation, Morrey spaces, and $\lambda$-central Carleson measures, Collect. Math., 51 (2000), 1–47. |
[4] | D. Cruz-Uribe, A. Fiorenza, C. Neugebauer, The maximal function on variable $L^p$ spaces, Ann. Acad. Sci. Fenn. Math., 28 (2003), 223–238. |
[5] | L. Diening, P. Harjulehto, P. Hästö, M. Růžička, Lebesgue and Sobolev spaces with variable exponents, Springer, 2011. http://doi.org/10.1007/978-3-642-18363-8 |
[6] | Z. Fu, $\lambda$-central BMO estimates for commutators of $n$-dimensional Hardy operators, J. Inequal. Pure Appl. Math., 9 (2008), 111. |
[7] | Z. Fu, S. Lu, H. Wang, L. Wang, Singular integral operators with rough kernels on central Morrey spaces with variable exponent, Ann. Acad. Sci. Fenn. Math., 44 (2019), 505–522. http://doi.org/10.5186/aasfm.2019.4431 doi: 10.5186/aasfm.2019.4431 |
[8] | Z. Fu, S. Lu, F. Zhao, Commutators of $n$-dimensional rough Hardy operators, Sci. China Math., 54 (2011), 95–104. http://doi.org/10.1007/s11425-010-4110-8 doi: 10.1007/s11425-010-4110-8 |
[9] | V. S. Guliyev, M. N. Omarova, M. A. Ragusa, A. Scapellato, Regularity of solutions of elliptic equations in divergence form in modified local generalized Morrey spaces, Anal. Math. Phys., 11 (2021), 13. http://doi.org/10.1007/s13324-020-00433-9 doi: 10.1007/s13324-020-00433-9 |
[10] | A. Hussain, M. Asim, M. Aslam, F. Jarad, Commutators of the fractional Hardy operator on weighted variable Herz-Morrey spaces, J. Funct. Spaces, 2021 (2021), 9705250. http://doi.org/10.1155/2021/9705250 doi: 10.1155/2021/9705250 |
[11] | A. Hussain, M. Asim, F. Jarad, Variable $\lambda$-central Morrey space estimates for the fractional Hardy operators and commutators, J. Math., 2022 (2022), 5855068. http://doi.org/10.1155/2022/5855068 doi: 10.1155/2022/5855068 |
[12] | M. Izuki, Boundedness of sublinear operators on Herz spaces with variable exponent and application to wavelet characterization, Anal. Math., 36 (2010), 33–50. http://doi.org/10.1007/s10476-010-0102-8 doi: 10.1007/s10476-010-0102-8 |
[13] | S. Khalid, J. Pečarić, Generalizations of some Hardy-Littlewood-Pólya type inequalities and related results, Filomat, 35 (2021), 2811–2826. http://doi.org/10.2298/FIL2108811K doi: 10.2298/FIL2108811K |
[14] | O. Kováčik, J. Rákosník, On spaces $L^{p(x)}$ and $W^{k, p(x)}$, Czech. Math. J., 41 (1991), 592–618. http://doi.org/10.21136/CMJ.1991.102493 doi: 10.21136/CMJ.1991.102493 |
[15] | Y. Mizuta, T. Ohno, T. Shimomura, Boundedness of maximal operators and Sobolev's theorem for non-homogeneous central Morrey spaces of variable exponent, Hokkaido Math. J., 44 (2015), 185–201. http://doi.org/10.14492/hokmj/1470053290 doi: 10.14492/hokmj/1470053290 |
[16] | E. Nakai, Y. Sawano, Hardy spaces with variable exponents and generalized Campanato spaces, J. Funct. Anal., 262 (2012), 3665–3748. http://doi.org/10.1016/j.jfa.2012.01.004 doi: 10.1016/j.jfa.2012.01.004 |
[17] | W. Orlicz, $\ddot{U}$ber konjugierte exponentenfolgen, Stud. Math., 3 (1931), 200–211. |
[18] | S. Shi, Z. Fu, S. Lu, On the compactness of commutators of Hardy operators, Pac. J. Math., 307 (2020), 239–256. http://doi.org/10.2140/pjm.2020.307.239 doi: 10.2140/pjm.2020.307.239 |
[19] | S. Shi, S. Lu, Characterization of the central Campanato space via the commutator operator of Hardy type, J. Math. Anal. Appl., 429 (2015), 713–732. http://doi.org/10.1016/j.jmaa.2015.03.083 doi: 10.1016/j.jmaa.2015.03.083 |
[20] | Z. Si, $\lambda$-central ${\rm{BMO}}$ estimates for multilinear commutators of fractional integrals, Acta Math. Sin. Engl. Ser., 26 (2010), 2093–2108. http://doi.org/10.1007/s10114-010-9363-1 doi: 10.1007/s10114-010-9363-1 |
[21] | A. Torchinsky, Real-variable methods in harmonic analysis, Academic Press, 1986. |
[22] | D. Wang, Z. Liu, J. Zhou, Z. Teng, Central BMO spaces with variable exponent, arXiv, 2017. https://doi.org/10.48550/arXiv.1708.00285 |
[23] | H. Wang, J. Xu, Multilinear fractional integral operators on central Morrey spaces with variable exponent, J. Inequal. Appl., 2019 (2019), 311. http://doi.org/10.1186/s13660-019-2264-7 doi: 10.1186/s13660-019-2264-7 |
[24] | H. Wang, J. Xu, J. Tan, Boundedness of multilinear singular integrals on central Morrey spaces with variable exponents, Front. Math. China, 15 (2020), 1011–1034. http://doi.org/10.1007/s11464-020-0864-7 doi: 10.1007/s11464-020-0864-7 |