Research Article
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Year 2024, Volume: 73 Issue: 3, 802 - 819, 27.09.2024

Muhammad Asim , Ferit Gürbüz

https://doi.org/10.31801/cfsuasmas.1463245

Abstract

References

  • Hardy, G. H., Note on a theorem of Hilbert, Math. Z., 6 (1920), 314-317.https://doi.org/10.1007/BF01199965
  • 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
  • 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
  • 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
  • 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.
  • 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
  • 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
  • Orlicz, W., Über konjugierte exponentenfolgen, Studia Math., 3(1931), 200-212. https://doi.org/10.4064/SM-3-1-200-211
  • Nakano, H., Modulared Semi-Ordered Linear Spaces, Maruzen Co, Ltd, Tokyo, 1951.
  • 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.
  • 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
  • Uribe, D. C., Fiorenza, A., Neugebauer, A., The maximal function on variable $L^{p}$ spaces, Ann. Acad. Sci. Fenn. Math., 28 (2003), 223-238.
  • 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
  • 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.
  • 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
  • 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
  • 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
  • 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
  • Ruicka, M., Electrorheogical Fluid: Modeling and Mathematical Theory, Springer, Berlin, 2000.
  • 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
  • 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
  • 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
  • Wang, D., Liu, Z., Zhou, J., Teng, Z., Central BMO spaces with variable exponent, arXiv:1708.00285, 2017.
  • 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
  • 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.
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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.
  • Grafakos, L., Modern Fourier Analysis , 2nd edition, Springer, 2009.
  • 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
  • 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

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

Year 2024, Volume: 73 Issue: 3, 802 - 819, 27.09.2024

Muhammad Asim , Ferit Gürbüz

https://doi.org/10.31801/cfsuasmas.1463245

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

  • Hardy, G. H., Note on a theorem of Hilbert, Math. Z., 6 (1920), 314-317.https://doi.org/10.1007/BF01199965
  • 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
  • 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
  • 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
  • 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.
  • 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
  • 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
  • Orlicz, W., Über konjugierte exponentenfolgen, Studia Math., 3(1931), 200-212. https://doi.org/10.4064/SM-3-1-200-211
  • Nakano, H., Modulared Semi-Ordered Linear Spaces, Maruzen Co, Ltd, Tokyo, 1951.
  • 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.
  • 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
  • Uribe, D. C., Fiorenza, A., Neugebauer, A., The maximal function on variable $L^{p}$ spaces, Ann. Acad. Sci. Fenn. Math., 28 (2003), 223-238.
  • 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
  • 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.
  • 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
  • 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
  • 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
  • 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
  • Ruicka, M., Electrorheogical Fluid: Modeling and Mathematical Theory, Springer, Berlin, 2000.
  • 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
  • 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
  • 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
  • Wang, D., Liu, Z., Zhou, J., Teng, Z., Central BMO spaces with variable exponent, arXiv:1708.00285, 2017.
  • 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
  • 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.
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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.
  • Grafakos, L., Modern Fourier Analysis , 2nd edition, Springer, 2009.
  • 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
  • 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
There are 37 citations in total.

Details

Primary Language English
Subjects Lie Groups, Harmonic and Fourier Analysis, Real and Complex Functions (Incl. Several Variables)
Journal Section Research Articles
Authors

Muhammad Asim 0000-0002-7336-9760

Ferit Gürbüz 0000-0003-3049-688X

Publication Date September 27, 2024
Submission Date April 2, 2024
Acceptance Date May 24, 2024
Published in Issue Year 2024 Volume: 73 Issue: 3

Cite

APA Asim, M., & Gürbüz, F. (2024). Some variable exponent boundedness and commutators estimates for fractional Rough Hardy operators on central Morrey space. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 73(3), 802-819. https://doi.org/10.31801/cfsuasmas.1463245
AMA Asim M, Gürbüz F. Some variable exponent boundedness and commutators estimates for fractional Rough Hardy operators on central Morrey space. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. September 2024;73(3):802-819. doi:10.31801/cfsuasmas.1463245
Chicago Asim, Muhammad, and Ferit Gürbüz. “Some Variable Exponent Boundedness and Commutators Estimates for Fractional Rough Hardy Operators on Central Morrey Space”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 73, no. 3 (September 2024): 802-19. https://doi.org/10.31801/cfsuasmas.1463245.
EndNote Asim M, Gürbüz F (September 1, 2024) Some variable exponent boundedness and commutators estimates for fractional Rough Hardy operators on central Morrey space. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 73 3 802–819.
IEEE M. Asim and F. Gürbüz, “Some variable exponent boundedness and commutators estimates for fractional Rough Hardy operators on central Morrey space”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 73, no. 3, pp. 802–819, 2024, doi: 10.31801/cfsuasmas.1463245.
ISNAD Asim, Muhammad - Gürbüz, Ferit. “Some Variable Exponent Boundedness and Commutators Estimates for Fractional Rough Hardy Operators on Central Morrey Space”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 73/3 (September 2024), 802-819. https://doi.org/10.31801/cfsuasmas.1463245.
JAMA Asim M, Gürbüz F. Some variable exponent boundedness and commutators estimates for fractional Rough Hardy operators on central Morrey space. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2024;73:802–819.
MLA Asim, Muhammad and Ferit Gürbüz. “Some Variable Exponent Boundedness and Commutators Estimates for Fractional Rough Hardy Operators on Central Morrey Space”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 73, no. 3, 2024, pp. 802-19, doi:10.31801/cfsuasmas.1463245.
Vancouver Asim M, Gürbüz F. Some variable exponent boundedness and commutators estimates for fractional Rough Hardy operators on central Morrey space. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2024;73(3):802-19.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.

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