International Journal of Analysis and Applications ISSN 2291-8639 (Online)

Research Article

On the Boundedness of Hardy Operators on λ-Central Amalgam-Morrey Spaces

  • Mehvish Sultan
  • Babar Sultan*
  • Ioan-Lucian Popa

    * Corresponding author

    Abstract

    In this paper, we introduce the idea of λ-central Amalgam-BMO spaces and λ-central Amalgam-Morrey spaces. We obtain the boundedness of the Hardy operators λ-central Amalgam-Morrey spaces.

    How to Cite

    M. Sultan, B. Sultan, I.-L. Popa, On the Boundedness of Hardy Operators on λ-Central Amalgam-Morrey Spaces, Int. J. Anal. Appl. 23 (2025), 320. https://doi.org/10.28924/2291-8639-23-2025-320

    BibTeX
    @article{Sultan2025,
      author  = {Sultan, Mehvish and Sultan, Babar and Popa, Ioan-Lucian},
      title   = {On the Boundedness of Hardy Operators on λ-Central Amalgam-Morrey Spaces},
      journal = {Int. J. Anal. Appl.},
      volume  = {23},
      year    = {2025},
      pages   = {320},
      doi     = {10.28924/2291-8639-23-2025-320}
    }

    References

    • [1] J. Alv\'arez, J. Lakey, M. Guzm\'an-Partida, Spaces of Bounded $\lambda$-Central Mean Oscillation, Morrey Spaces, and $\lambda$-Central Carleson Measures, Collect. Math. 51 (2000), 1–47. https://eudml.org/doc/41472.
    • [2] S. Baena-Miret, Weighted Strong-Type Estimates on Classical Lorentz Spaces, J. Geom. Anal. 34 (2024), 86. https://doi.org/10.1007/s12220-023-01519-z.
    • [3] J. Bertrandias, C. Datry, C. Dupuis, Unions et Intersections d'Espaces $L^p$ Invariantes par Translation ou Convolution, Ann. Inst. Fourier 28 (1978), 53–84. https://doi.org/10.5802/aif.689.
    • [4] Y. Chen, S. Levine, M. Rao, Variable Exponent, Linear Growth Functionals in Image Restoration, SIAM J. Appl. Math. 66 (2006), 1383–1406. https://doi.org/10.1137/050624522.
    • [5] M. Christ, L. Grafakos, Best Constants for Two Nonconvolution Inequalities, Proc. Am. Math. Soc. 123 (1995), 1687–1693. https://doi.org/10.1090/s0002-9939-1995-1239796-6.
    • [6] M. Cowling, S. Meda, R. Pasquale, Riesz Potentials and Amalgams, Ann. Inst. Fourier 49 (1999), 1345–1367. https://doi.org/10.5802/aif.1720.
    • [7] J.J.F. Fournier, On the Hausdorff-Young Theorem for Amalgams, Monatsh. Math. 95 (1983), 117–135. https://doi.org/10.1007/bf01323655.
    • [8] Z. Fu, Z. Liu, S. Lu, H. Wang, Characterization for Commutators of N-Dimensional Fractional Hardy Operators, Sci. China Ser. A: Math. 50 (2007), 1418–1426. https://doi.org/10.1007/s11425-007-0094-4.
    • [9] Z. Fu, S. Lu, F. Zhao, Commutators of N-Dimensional Rough Hardy Operators, Sci. China Math. 54 (2011), 95–104. https://doi.org/10.1007/s11425-010-4110-8.
    • [10] 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. https://doi.org/10.5186/aasfm.2019.4431.
    • [11] Z.W. Fu, Y. Lin, S.Z. Lu, $\lambda$-Central BMO Estimates for Commutators of Singular Integral Operators with Rough Kernels, Acta Math. Sin. Engl. Ser. 24 (2008), 373–386. https://doi.org/10.1007/s10114-007-1020-y.
    • [12] G.H. Hardy, Note on a Theorem of Hilbert, Math. Z. 6 (1920), 314–317. https://doi.org/10.1007/bf01199965.
    • [13] Y. Lu, S. Wang, J. Zhou, Boundedness of Some Operators on Weighted Amalgam Spaces, arXiv:2110.01193 (2021). https://doi.org/10.48550/arXiv.2110.01193.
    • [14] 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. https://doi.org/10.14492/hokmj/1470053290.
    • [15] C.B. Morrey, On the Solutions of Quasi-Linear Elliptic Partial Differential Equations, Trans. Am. Math. Soc. 43 (1938), 126–166. https://doi.org/10.2307/1989904.
    • [16] C. Niu, H. Wang, Hardy-Type Operators with Rough Kernels on Central Morrey Space with Variable Exponent, Adv. Oper. Theory 8 (2023), 26. https://doi.org/10.1007/s43036-023-00246-0.
    • [17] B. Sultan, M. Sultan, Boundedness of Commutators of Rough Hardy Operators on Grand Variable Herz Spaces, Forum Math. 36 (2023), 717–733. https://doi.org/10.1515/forum-2023-0152.
    • [18] B. Sultan, M. Sultan, Boundedness of Higher Order Commutators of Hardy Operators on Grand Herz-Morrey Spaces, Bull. Sci. Math. 190 (2024), 103373. https://doi.org/10.1016/j.bulsci.2023.103373.
    • [19] B. Sultan, M. Sultan, Q. Zhang, N. Mlaiki, Boundedness of Hardy Operators on Grand Variable Weighted Herz Spaces, AIMS Math. 8 (2023), 24515–24527. https://doi.org/10.3934/math.20231250.
    • [20] M. Sultan, B. Sultan, Boundedness of Sublinear Operators on Grand Central Orlicz-Morrey Spaces, Bull. Sci. Math. 205 (2025), 103704. https://doi.org/10.1016/j.bulsci.2025.103704.
    • [21] M. Sultan, B. Sultan, $\lambda$-Central Musielak–Orlicz–Morrey Spaces, Arab. J. Math. 14 (2025), 357–363. https://doi.org/10.1007/s40065-025-00528-w.
    • [22] B. Sultan, M. Sultan, I. Khan, On Sobolev Theorem for Higher Commutators of Fractional Integrals in Grand Variable Herz Spaces, Commun. Nonlinear Sci. Numer. Simul. 126 (2023), 107464. https://doi.org/10.1016/j.cnsns.2023.107464.
    • [23] B. Sultan, M. Sultan, Sobolev-Type Theorem for Commutators of Hardy Operators in Grand Herz Spaces, Ukr. Math. J. 76 (2024), 1196–1213. https://doi.org/10.1007/s11253-024-02381-0.
    • [24] M. Sultan, B. Sultan, R.E. Castillo, Weighted Composition Operator on Gamma Spaces with Variable Exponent, J. Pseudo-Differential Oper. Appl. 15 (2024), 46. https://doi.org/10.1007/s11868-024-00619-w.
    • [25] M. Sultan, B. Sultan, A Note on the Boundedness of Higher Order Commutators of Fractional Integrals in Grand Variable Herz-Morrey Spaces, Kragujev. J. Math. 50 (2026), 1063–1080.
    • [26] M. Sultan, B. Sultan, A Note on the Boundedness of Marcinkiewicz Integral Operator on Continual Herz-Morrey Spaces, Filomat 39 (2025), 2017–2027. https://doi.org/10.2298/fil2506017s.
    • [27] Y. Sawano, T. Shimomura, Boundedness of the Generalized Fractional Integral Operators on Generalized Morrey Spaces Over Metric Measure Spaces, Z. Anal. Anwend. 36 (2017), 159–190. https://doi.org/10.4171/zaa/1584.
    • [28] Z.Y. Si, $\lambda$-Central BMO Estimates for Multilinear Commutators of Fractional Integrals, Acta Math. Sin. Engl. Ser. 26 (2010), 2093–2108. https://doi.org/10.1007/s10114-010-9363-1.
    • [29] M. Sultan, B. Sultan, R.E. Castillo, Lorentz Herz-Morrey Spaces with Applications, J. Pseudo-Differ. Oper. Appl. 16 (2025), 55. https://doi.org/10.1007/s11868-025-00697-4.
    • [30] A. Hussain, I. Khan, A. Mohamed, Variable Herz–Morrey Estimates for Rough Fractional Hausdorff Operator, J. Inequal. Appl. 2024 (2024), 33. https://doi.org/10.1186/s13660-024-03110-8.
    • [31] A. Ajaib, A. Hussain, Weighted CBMO Estimates for Commutators of Matrix Hausdorff Operator on the Heisenberg Group, Open Math. 18 (2020), 496–511. https://doi.org/10.1515/math-2020-0175.
    • [32] J. Younas, A. Hussain, H. Alhazmi, A.F. Aljohani, I. Khan, BMO Estimates for Commutators of the Rough Fractional Hausdorff Operator on Grand-Variable-Herz-Morrey Spaces, AIMS Math. 9 (2024), 23434–23448. https://doi.org/10.3934/math.20241139.
    • [33] H. Wang, J. Xu, Multilinear Fractional Integral Operators on Central Morrey Spaces with Variable Exponent, J. Inequal. Appl. 2019 (2019), 311. https://doi.org/10.1186/s13660-019-2264-7.
    • [34] N. Wiener, On the Representation of Functions by Trigonometrical Integrals, Math. Z. 24 (1926), 575–616. https://doi.org/10.1007/bf01216799.