On the Boundedness of Hausdorff Operators in Block Space with Variable Exponent

Sultan, Mehvish; Sultan, Babar; Popa, Ioan-Lucian

doi:10.28924/2291-8639-23-2025-226

Research Article

On the Boundedness of Hausdorff Operators in Block Space with Variable Exponent

Mehvish Sultan
Babar Sultan*
Ioan-Lucian Popa

* Corresponding author

Abstract

In this paper, we prove the Hardy inequality, the Hilbert inequality and the Hardy-Littlewood-Pólya inequality on block spaces with variable exponents. Furthermore, we establish the boundedness of Hausdorff operators on block space with variable exponent.

How to Cite

M. Sultan, B. Sultan, I.-L. Popa, On the Boundedness of Hausdorff Operators in Block Space with Variable Exponent, Int. J. Anal. Appl. 23 (2025), 226. https://doi.org/10.28924/2291-8639-23-2025-226

BibTeX

@article{Sultan2025,
  author  = {Sultan, Mehvish and Sultan, Babar and Popa, Ioan-Lucian},
  title   = {On the Boundedness of Hausdorff Operators in Block Space with Variable Exponent},
  journal = {Int. J. Anal. Appl.},
  volume  = {23},
  year    = {2025},
  pages   = {226},
  doi     = {10.28924/2291-8639-23-2025-226}
}

References

[1] O. Blasco, A. Ruiz, L. Vega, Non Interpolation in Morrey-Campanato and Block Spaces, Ann. Sc. Norm. Super. Pisa, Cl. Sci. 28 (1999), 31–40. http://eudml.org/doc/84372.
[2] D.E. Edmunds, W.D. Evans, Hardy Operators, Function Spaces and Embeddings, Springer Berlin, Berlin, 2004. https://doi.org/10.1007/978-3-662-07731-3.
[3] G. Hardy, J. Littlewood, G. P\"olya, Inequalities, Cambridge University Press, 1988.
[4] Y. Kanjin, The Hausdorff Operators on the Real Hardy Spaces $H^p(R)$, Stud. Math. 148 (2001), 37–45. https://doi.org/10.4064/sm148-1-4.
[5] A.K. Lerner, E. Liflyand, Multidimensional Hausdorff Operators on the Real Hardy Space, J. Aust. Math. Soc. 83 (2007), 79–86. https://doi.org/10.1017/s1446788700036399.
[6] E. Liflyand, Boundedness of Multidimensional Hausdorff Operators on $H^1(\mathbb{R}^n)$, Acta Sci. Math. (Szeged) 74 (2008), 845–851. https://cir.nii.ac.jp/crid/1370289302291212419.
[7] E. Liflyand, Hausdorff operators on Hardy spaces, Eurasian Math. J. 4 (2013), 101–141.
[8] E. Liflyand, F. M\'oricz, The Hausdorff Operators Is Bounded on the Real Hardy Space $H^1(\mathbb{R}^n)$, Proc. Amer. Math. Soc. 128 (2000), 1391–1396. https://doi.org/10.1090/s0002-9939-99-05159-x.
[9] E. Liflyand, F. M\'oricz, The Multi-Parameter Hausdorff Operator Is Bounded on the Product Hardy Space $H^{11}(\mathbb{R}\times \mathbb{R})$, Analysis 21 (2001), 107–118. https://doi.org/10.1524/anly.2001.21.2.107.
[10] E. Liflyand, A. Miyachi, Boundedness of the Hausdorff Operators in $H^p$ Spaces, $0 < p < 1$, Stud. Math. 194 (2009), 279–292. https://doi.org/10.4064/sm194-3-4.
[11] 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.
[12] B. Opic, A. Kufner, Hardy-Type Inequalities, Longman Scientific \& Technical, 1990.
[13] 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.
[14] M. Sultan, B. Sultan, A. Khan, T. Abdeljawad, Boundedness of Marcinkiewicz Integral Operator of Variable Order in Grand Herz-Morrey Spaces, AIMS Math. 8 (2023), 22338–22353. https://doi.org/10.3934/math.20231139.
[15] 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.
[16] M. Sultan, B. Sultan, A. Hussain, Grand Herz–morrey Spaces with Variable Exponent, Math. Notes 114 (2023), 957–977. https://doi.org/10.1134/s0001434623110305.
[17] M. Sultan, B. Sultan, A. Aloqaily, N. Mlaiki, Boundedness of Some Operators on Grand Herz Spaces with Variable Exponent, AIMS Math. 8 (2023), 12964–12985. https://doi.org/10.3934/math.2023653.
[18] B. Sultan, F. Azmi, M. Sultan, M. Mehmood, N. Mlaiki, Boundedness of Riesz Potential Operator on Grand Herz-Morrey Spaces, Axioms 11 (2022), 583. https://doi.org/10.3390/axioms11110583.
[19] 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.
[20] A. Hussain, G. Gao, Multilinear Singular Integrals and Commutators on Herz Space with Variable Exponent, ISRN Math. Anal. 2014 (2014), 626327. https://doi.org/10.1155/2014/626327.
[21] 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.
[22] 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.
[23] 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.
[24] B. Sultan, M. Sultan, Sobolev-type Theorem for Commutators of Hardy Operators in Grand Herz Spaces, Ukr. Mat. Zhurnal 76 (2024), 1052–1068. https://doi.org/10.3842/umzh.v76i7.7546.
[25] 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.
[26] M. Sultan, B. Sultan, A Note on the Boundedness of Higher Order Commutators on Fractional Integrals in Grand Variable Herz-Morrey Spaces, Kragujev. J. Math. 50 (2026), 1063–1080.
[27] B. Sultan, M. Sultan, A. Khan, T. Abdeljawad, Boundedness of Commutators of Variable Marcinkiewicz Fractional Integral Operator in Grand Variable Herz Spaces, J. Inequal. Appl. 2024 (2024), 93. https://doi.org/10.1186/s13660-024-03169-3.
[28] 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.
[29] B. Sultan, M. Sultan, A. Hussain, Boundedness of the Bochner–Riesz Operators on the Weighted Herz–Morrey Type Hardy Spaces, Complex Anal. Oper. Theory 19 (2025), 49. https://doi.org/10.1007/s11785-025-01671-0.
[30] B. Sultan, A. Hussain, M. Sultan, Chracterization of Generalized Campanato Spaces with Variable Exponents via Fractional Integrals, J. Pseudo-Differential Oper. Appl. 16 (2025), 22. https://doi.org/10.1007/s11868-024-00651-w.
[31] B. Sultan, M. Sultan, M. Mehmood, F. Azmi, M.A. Alghafli, N. Mlaiki, Boundedness of Fractional Integrals on Grand Weighted Herz Spaces with Variable Exponent, AIMS Math. 8 (2023), 752–764. https://doi.org/10.3934/math.2023036.
[32] B. Sultan, F.M. Azmi, M. Sultan, T. Mahmood, N. Mlaiki, N. Souayah, Boundedness of Fractional Integrals on Grand Weighted Herz–morrey Spaces with Variable Exponent, Fractal Fract. 6 (2022), 660. https://doi.org/10.3390/fractalfract6110660.
[33] 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.
[34] 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.