Necessary and sufficient conditions for the boundedness of some discrete operators on Morrey sequence spaces

Abstract

The aim of this paper is to study the boundedness properties of the discrete Riesz potential, the fractional maximal operator, and their commutators on Morrey sequence spaces. We establish both necessary and sufficient conditions for the boundedness of the operators in the discrete Morrey framework. We also introduce the absolute higher-order commutators of the discrete Riesz potential and the fractional maximal operator. Moreover, we derive discrete analogues of the John--Nirenberg and Fefferman--Stein inequalities and employ them to prove the boundedness of the absolute higher-order commutators on Morrey sequence spaces. In addition, a sufficient condition for the boundedness of the absolute higher-order commutators is obtained. As a consequence, we deduce the boundedness of the higher-order commutators of the discrete Riesz potential and the fractional maximal operator on Morrey sequence spaces, thereby extending several classical results from the continuous to the discrete setting.

Keywords

• Riesz potential
• fractional maximal operator
• commutator
• Lebesgue space
• Morrey space
• sequence space

References

1. [1] I. Gabisonija and A. Meskhi, Two-weighted inequalities for a discrete Hilbert transform, Proc. A. Razmadze Math. Inst. 116, 107-122, 1998.
2. [2] J. Garcia-Cuerva and J.L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, Elsevier Science Publishers B.V., Amsterdam, 1985.
3. [3] H. Gunawan, D.I. Hakim, Ifronika and O. Neswan, Inclusion and geometric properties of mixed Morrey double-sequence spaces, Bull. Malays. Math. Sci. Soc. 47, 132, 2024.
4. [4] H. Gunawan, E. Kikianty and C. Schwanke, Discrete Morrey spaces and their inclusion properties, Math. Nachr. 291 (8-9), 1283-1296, 2018.
5. [5] H. Gunawan and C. Schwanke, The Hardy-Littlewood maximal operator on discrete Morrey spaces, Mediterr. J. Math. 16 (1), 1-12, 2019.
6. [6] X.B. Hao, S. Yang and B. Li, Estimates of discrete Riesz potentials on discrete weighted Lebesgue spaces, Ann. Funct. Anal. 15, 2024.
7. [7] X.B. Hao, S. Yang and B. Li, The Hardy-Littlewood Maximal Operator on Discrete Weighted Morrey Spaces, Acta Math. Hungar. 172, 445-469, 2024.
8. [8] D.D. Haroske and L. Skrzypczak, Morrey sequence spaces: Pitts theorem and compact embeddings, Constr. Approx. 51, 505-535, 2020.

9. [9] D.D. Haroske and L. Skrzypczak, Generalised Morrey sequence spaces, Adv. Oper. Theory 10, 2025.
10. [10] M. Jakfar, M. Manuharawati, A. Lukito and S. Fiangga, Inner products on discrete Morrey spaces, Eur. J. Pure Appl. Math. 16 (1), 144-155, 2023.
11. [11] S. Janson, Mean oscillation and commutators of singular integral operators, Ark. Mat. 16, 263-270, 1978.
12. [12] C.B. Morrey, On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc. 43 (1), 126-166, 1938.
13. [13] Y. Ramadana, Some operators on generalized discrete Morrey spaces over $\mathbb{Z}^d$, Complex Anal. Oper. Theory 19, 2025.
14. [14] Y.I. Runtunuwu, J. Lindiarni and H. Gunawan, On the proper inclusion property of (discrete) Morrey spaces, Czech Math. J. 75, 955-961, 2025.