Abstract
We introduce a discrete version of weighted Morrey spaces, and discuss the inclusion relations of these spaces. In addition, we obtain the boundedness of discrete weighted Hardy-Littlewood maximal operators on discrete weighted Lebesgue spaces by establishing a discrete Calderón-Zygmund decomposition for weighted \(l^1\)-sequences. Furthermore, the necessary and sufficient conditions for the boundedness of the discrete Hardy-Littlewood maximal operators on discrete weighted Morrey spaces are discussed. Particularly, the necessary and sufficient conditions are also discussed for the discrete power weights.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Y. Abe and Y. Sawano, Littlewood-Paley characterization of discrete Morrey spaces and its application to the discrete martingale transform, Matematiche (Catania), to appear.
R. A. Aliev, A. N. Ahmadova and A. F. Huseynli, Boundedness of the discrete Ahlfors- Beurling transform on discrete Morrey spaces, Inst. Math. Mech., 48 (2022), 123-131.
A. R. Avazaga and A. A. Nofel, Boundedness of discrete Hilbert transform on discrete Morrey spaces, Ufa Math. J., 13 (2021), 98-109.
A. Böttcher and M. Seybold, Discrete Wiener-Hopf operators on spaces with Muckenhoupt weight, Studia Math., 143 (2000), 121-144.
F. Chiarenza and M. Frasca, Morrey spaces and Hardy-Littlewood maximal function, Rend. Mat. Appl., 7 (1987), 273-279.
J. Duoandikoetxea, Fourier Analysis, American Mathematical Society (Providence, RI, 2001).
J. Duoandikoetxea and M. Rosenthal, Boundedness properties in a family of weighted Morrey spaces with emphasis on power weights, J. Funct. Anal., 279 (2020), 108687, 26 pp.
X. Fu, Weighted boundedness of discrete fractional integrals, Sci. Sin. Math., 51 (2021), 333-342.
J. Garcia-Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, Elsevier (New York, 2011).
H. Gunawan, E. Kikianty and C. Schwanke, Discrete Morrey spaces and their inclusion properties, Math. Nachr., 291 (2018), 1283-1296.
H. Gunawan and C. Schwanke, The Hardy-Littlewood maximal operator on discrete Morrey spaces, Mediterr. J. Math., 16 (2019), Paper No. 24, 12 pp.
D. D. Haroske and L. Skrzypczak, Morrey sequence spaces: Pitt's theorem and compact embeddings, Constr. Approx., 51 (2020), 505-535.
E. Kikianty and C. Schwanke, Discrete Morrey spaces are closed subspaces of their continuous counterparts, Banach Center Publ., 119 (2019), 223-231.
Y. Komori and S. Shirai, Weighted Morrey spaces and a singular integral operator, Math. Nachr., 282 (2009), 219-231.
S. Lu, Y. Ding and D. Yan, Singular Integrals and Related Topics, World Scientific (Singapore, 2007).
C. B. Morrey, On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc., 43 (1938), 126-166.
S. Nakamura, Y. Sawano and H. Tanaka, The fractional operators on weighted Morrey spaces, J. Geom. Anal., 28 (2018), 1502-1524.
S. H. Saker and R. P. Agarwal, Theory of discrete Muckenhoupt weights and discrete Rubio de Francia extrapolation theorems, Appl. Anal. Discrete Math., 15 (2021), 295-316.
S. H. Saker and I. Kubiaczyk, Higher summability and discrete weighted Muckenhoupt and Gehring type inequalities, Proc. Edinb. Math. Soc., 62 (2019), 949-973.
S. H. Saker and R. R. Mahmoud, Boundedness of both discrete Hardy and Hardy- Littlewood maximal operators via Muckenhoupt weights, Rocky Mountain J. Math., 51 (2021), 733-746.
S. H. Saker, S. S. Rabie and J. Alzabut, D. O'Regan and R. P. Agarwal, Some basic properties and fundamental relations for discrete Muckenhoupt and Gehring classes, Adv. Difference Equ. (2021), Paper No. 8, 22 pp.
Y. Sawano, G. D. Fazio and D. I. Hakim, Morrey Spaces: Introduction and Applications to Integral Operators and PDE's. Vols. I,II, CRC Press (Boca Raton, FL, 2020).
A. S. Swarup and A. M. Alphonse, The boundedness of fractional Hardy-Littlewood maximal operator on variable \(l^p\)spaces using Calderón-Zygmund decomposition, arXiv: 2204.04331 (2022).
H. Tanaka, Two-weight norm inequalities on Morrey spaces, Ann. Acad. Sci. Fenn. Math., 40 (2015), 773-791.
D. H. Wang, J. Zhou and W. Y. Chen, Another characterizations of Muckenhoupt \(A_p\) class, Acta Math. Sci., 37 (2017), 1761-1774.
M. Wei and X. Liu, Sharp weak bounds for discrete Hardy operator on discrete central Morrey spaces, AIMS Math., 8 (2023), 5007-5015.
Acknowledgement
The authors express their deep gratitude to the referees for their meticulous work and useful comments which do improve Theorems 4.6 and 4.12, and the presentation of this article.
Author information
Authors and Affiliations
Corresponding author
Additional information
Baode Li is supported by the National Natural Science Foundation of China (Grant No. 12261083).
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Hao, X.B., Li, B.D. & Yang, S. The Hardy–Littlewood maximal operator on discrete weighted Morrey spaces. Acta Math. Hungar. 172, 445–469 (2024). https://doi.org/10.1007/s10474-024-01420-3
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10474-024-01420-3
Key words and phrases
- weight
- discrete Morrey space
- discrete Hardy–Littlewood maximal operator
- discrete Calderón–Zygmund decomposition