Marcinkiewicz-type interpolation theorem for discrete net spaces
DOI:
https://doi.org/10.31489/2026m2/80-89Keywords:
net spaces, discrete net spaces, Lorentz space, Marcinkiewicz-type interpolation theorem, real interpolation method, linear operators, local nets, global netsAbstract
In this paper, we investigate the interpolation properties of discrete net spaces np,q(M) and examine their applications to the analysis of linear operators acting on these spaces. These spaces are characterized by the property that, for monotonically non-increasing sequences, the norm in np,q(M) coincides with the norm of the discrete Lorentz space lp,q(M). At the same time, unlike Lorentz spaces, these spaces np,q(M) may contain sequences that do not tend to zero, making them suitable for the study of more general function spaces and operator classes. The main result of this paper is an analogue of Marcinkiewicz-type interpolation theorem for discrete net spaces np,q(M), which offers a powerful tool to study the boundedness of linear operators within this framework. By extending classical interpolation techniques to discrete nets, the theorem enables researchers to derive strong-type estimates for operators based on weak-type estimates on local nets. Consequently, this approach provides a unified framework for obtaining boundedness results, demonstrating the utility of discrete net spaces in analyzing operators within harmonic analysis. These findings contribute significantly to understanding the structural properties of discrete net spaces. Furthermore, they introduce innovative tools for applications in harmonic analysis, operator theory, and related mathematical fields where such spaces naturally arise, ultimately paving the way for advanced theoretical developments and broader analytical applications.
References
Nursultanov, E.D. (1998). Net spaces and inequalities of Hardy–Littlewood type. Sbornik: Mathematics — Collection: Mathematics: 189(3), 399–419. https://doi.org/10.1070/sm1998v189n03ABEH000309 DOI: https://doi.org/10.1070/SM1998v189n03ABEH000309
Bergh, J., & L¨ofstr¨om, J. (1976). Interpolation Spaces: An Introduction. Springer. https://doi.org/10.1007/978-3-642-66451-9 DOI: https://doi.org/10.1007/978-3-642-66451-9_5
Triebel, H. (2010). Theory of Function Spaces (2nd ed.). Birkh¨auser.
Akylzhanov, R., & Ruzhansky, M. (2020). Lp−Lq multipliers on locally compact groups. Journal of Functional Analysis, 278(3), Article 108324. https://doi.org/10.1016/j.jfa.2019.108324 DOI: https://doi.org/10.1016/j.jfa.2019.108324
Akylzhanov, R., & Ruzhansky, M. (2017). Net spaces on lattices, Hardy–Littlewood type inequalities, and their converses. Eurasian Mathematical Journal, 8(3), 10–27. http://hdl.handle.net/1854/LU-8585434
Mukanov, A.B. (2015). On the general net spaces. Journal of Mathematics, Mechanics and Computer Science, 87(4), 27–34.
Suragan, D., & Zaighum, M. (2025). Grand net spaces and applications to integral operators. arXiv:2503.20080. https://doi.org/10.48550/arXiv.2503.20080 DOI: https://doi.org/10.1007/s12215-026-01429-3
Kokilashvili, V., Meskhi, A., Rafeiro, H., & Samko, S. (2016). Integral operators in non-standard function spaces. Volume 1: Variable exponent Lebesgue and amalgam spaces. Operator Theory: Advances and Applications (Vol. 248). Basel: Birkhauser. DOI: https://doi.org/10.1007/978-3-319-21015-5_1
Kokilashvili, V., Meskhi, A., Rafeiro, H., & Samko, S. (2016). Integral operators in non-standard function spaces. Volume 2: Variable exponent Holder, Morrey-Campanato and grand spaces. Operator Theory: Advances and Applications (Vol. 249). Basel: Birkhauser. DOI: https://doi.org/10.1007/978-3-319-21018-6
Kokilashvili, V., Meskhi, A., Rafeiro, H., & Samko, S. (2024). Integral operators in non-standard function spaces. Volume 3: Advances in grand function spaces. Operator Theory: Advances and Applications (Vol. 298). Basel: Birkhauser. DOI: https://doi.org/10.1007/978-3-031-64983-7
Tleukhanova, N.T., & Bakhyt, A. (2021). On trigonometric Fourier series multipliers in λp,q spaces. Eurasian Mathematical Journal, 12(1), 103–106. https://doi.org/10.32523/2077-9879-2021-12-1-103-106 DOI: https://doi.org/10.32523/2077-9879-2021-12-1-103-106
Tleukhanova, N.T., & Sadykova, K.K. (2019). O’Neil-type inequalities for convolutions in anisotropic Lorentz spaces. Eurasian Mathematical Journal, 10(3), 68–83. https://doi.org/10.32523/2077-9879-2019-10-3-68-83 DOI: https://doi.org/10.32523/2077-9879-2019-10-3-68-83
Sette, R.M.M., Fernandez, D.L., & da Silva, E.B. (2024). A Theory for Interpolation of Metric Spaces. Axioms, 13(7), Article 439. https://doi.org/10.3390/axioms13070439 DOI: https://doi.org/10.3390/axioms13070439
Tanaka, H. (2010). Morrey spaces and fractional operators. Journal of the Australian Mathematical Society, 88, 247–259. https://api.semanticscholar.org/CorpusID:121909948 DOI: https://doi.org/10.1017/S1446788709000457
Burenkov, V.I., & Goldman, M.L. (2014). Necessary and sufficient conditions for the boundedness of the maximal operator from Lebesgue spaces to Morrey-type spaces. Mathematical Inequalities & Applications, 17(2), 401–418. https://doi.org/10.7153/mia-17-30 DOI: https://doi.org/10.7153/mia-17-30
Yudatama, R., & Hakim, D.I. (2025). Intermediate spaces on weak type discrete Morrey spaces. Hilbert Journal of Mathematical Analysis, 3(2), 012–018. https://doi.org/10.62918/hjma.v3i2.33 DOI: https://doi.org/10.62918/hjma.v3i2.33
Zhang, F., & Zhou, J. (2025). Generalized Local Morrey Spaces Associated with Ball Banach Function Spaces and Their Application. Axioms, 14(12), Article 894. https://doi.org/10.3390/axioms14120894 DOI: https://doi.org/10.3390/axioms14120894
Ruiz, A., & Vega, L. (1995). Corrigenda to “Unique continuation for Schrodinger operator” and a remark on interpolation of Morrey spaces. Publicacions Matem`atiques, 39, 405–411. DOI: https://doi.org/10.5565/PUBLMAT_39295_15
Blasco, O., Ruiz, A., & Vega, L. (1999). Non interpolation in Morrey-Campanato and block spaces. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, 4e serie, 28(1), 31–40.
Hakim, D. (2020). Calderon’s complex interpolation of Morrey spaces. Journal of the Indonesian Mathematical Society, 26(1), 1–22. https://doi.org/10.22342/jims.26.1.818.137-164 DOI: https://doi.org/10.22342/jims.26.1.818.137-164
Zhuo, C., Hovemann, M., & Sickel, W. (2020). Complex interpolation of Lizorkin-TriebelMorrey spaces on domains. Analysis and Geometry in Metric Spaces, 8(1), 268–304. https://doi.org/10.1515/agms-2020-0114 DOI: https://doi.org/10.1515/agms-2020-0114
Burenkov, V.I., Chigambayeva, D.K., & Nursultanov, E.D. (2018). Marcinkiewicz-type interpolation theorem and estimates for convolutions for Morrey-type spaces. Eurasian Mathematical Journal, 9(2), 82–88. https://emj.enu.kz/index.php/main/article/view/118 DOI: https://doi.org/10.32523/2077-9879-2018-9-2-82-88
Chigambayeva, D. (2024). Marcinkiewicz-type interpolation theorem for local Morrey spaces. Bulletin of Abai Kazakh National Pedagogical University. Series: Physical and Mathematical Sciences, 88(4), 63–70.
Kalidolday, A.K., & Nursultanov, E.D. (2022). Marcinkiewicz’s interpolation theorem for linear operators on net spaces. Eurasian Mathematical Journal, 13(4), 61–69. https://doi.org/10.32523/2077-9879-2022-13-4-61-69 DOI: https://doi.org/10.32523/2077-9879-2022-13-4-61-69
Kalidolday, A.K., & Nursultanov, E.D. (2023). Interpolation theorem for discrete net spaces. Journal of Mathematics, Mechanics and Computer Science, 120(4), 24–31. https://doi.org/10.26577/JMMCS2023v120i4a3 DOI: https://doi.org/10.26577/JMMCS2023v120i4a3
