On the boundedness of Riesz potential operators: insights from net spaces

Erlan Nursultanov; Durvudkhan Suragan; Muhammad Asad Zaighum

doi:10.33205/cma.1716123

On the boundedness of Riesz potential operators: insights from net spaces

Abstract

This paper investigates the boundedness of Riesz potential operators on net spaces that are structured around special nets. We use the construction of net spaces and their intrinsic properties to establish conditions under which the considered operators are bounded. The methodology developed here provides a framework for establishing Hardy–Littlewood–Sobolev inequalities on net spaces, aiming at a deeper understanding of potential theory in non-standard settings.

Keywords

References

D. R. Adams: A note on Riesz potentials, Duke Math. J., 42 (4) (1975), 765–778.
S. V. Astashkin, L. Maligranda: Structure of Cesàro function spaces: a survey, in Function spaces X, 13–40, Banach Center Publ., 102, Polish Acad. Sci. Inst. Math., Warsaw (2014).
A. N. Bashirova, A. H. Kalidolday and E. D. Nursultanov: Interpolation methods for anisotropic net spaces, Eurasian Math. J., 15 (2) (2024), 33–41.
A. N. Bashirova, E. D. Nursultanov: The Hardy-Littlewood theorem for double Fourier-Haar series from mixed metric Lebesgue Lp[0, 1]2 and net Np,q(M) spaces, Anal. Math., 48(1) (2022), 5–17.
C. Bennett, R. C. Sharpley: Interpolation of operators, Pure and Applied Mathematics, 129, Academic Press, Boston, MA (1988).
R. E. Castillo, H. Rafeiro: An introductory course in Lebesgue spaces, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, Springer, Cham (2016).
L. Diening: Maximal function on generalized Lebesgue spaces Lp(·), Math. Inequal. Appl., 7 (2) (2004), 245–253.
L. Diening: Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spaces Lp(·) andWk,p(·), Math. Nachr., 268 (2004), 31–43.

G. H. Hardy, J. E. Littlewood: Some properties of fractional integrals. I, Math. Z., 27 (1928), 565–606.
A. H. Kalidolday, E. D. Nursultanov: Interpolation properties of certain classes of net spaces, Lobachevskii J. Math., 44 (5) (2023), 1870–1878.
A. H. Kalidolday, E. D. Nursultanov: Marcinkiewicz’s interpolation theorem for linear operators on net spaces, Eurasian Math. J., 13 (4) (2022), 61–69.
V. Kokilashvili, A. Meskhi, H. Rafeiro and S. Samko: Integral operators in non-standard function spaces. Vol. 1, Operator Theory: Advances and Applications, 248, Birkhäuser/Springer, Cham (2016).
V. Kokilashvili, A. Meskhi, H. Rafeiro and S. Samko: Integral operators in non-standard function spaces. Vol. 2, Operator Theory: Advances and Applications, 249, Birkhäuser/Springer, Cham (2016).
V. Kokilashvili, A. Meskhi, H. Rafeiro and S. Samko: Integral operators in non-standard function spaces. Vol. 3. Advances in grand function spaces, Operator Theory: Advances and Applications, 298, Birkhäuser/Springer, Cham (2024).
P. I. Lizorkin: Operators connected with fractional differentiation, and classes of differentiable functions, Trudy Mat. Inst. Steklov., 117 (1972), 212–243.
A. B. Mukanov: On the general net spaces, Journal of Mathematics, Mechanics and Computer Science, 87 (4) (2015), 27–34.
R. O’ Neil: Convolution operators and L(p,q) spaces, Duke Math. J., 30 (1) (1963), 129–142.
E. D. Nursultanov: Net spaces and inequalities of Hardy-Littlewood type, Sb. Math., 189 (3–4) (1998), 399–419.
E. D. Nursultanov and S. Y. Tikhonov: Net spaces and boundedness of integral operators, J. Geom. Anal., 21 (4) (2011), 950–981.
J. Peetre: On the theory of Lp,λ spaces, J. Functional Analysis, 4 (1969), 71–87.
S. G. Samko: Convolution and potential type operators in Lp(x)(Rn), Integral Transform. Spec. Funct., 7 (3–4) (1998), 261–284.
S. L. Sobolev: On a theorem of functional analysis, Sb. Math., 46 (4) (1938), 471–497.
E. M. Stein: Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton Univ. Press, Princeton, NJ (1970).
A. Torchinsky: Real-variable methods in harmonic analysis, Pure and Applied Mathematics, 123, Academic Press, Orlando, FL (1986).

Details

Primary Language

English

Subjects

Lie Groups, Harmonic and Fourier Analysis

Journal Section

Research Article

Authors

Erlan Nursultanov This is me
0000-0003-3879-2261
Kazakhstan

Durvudkhan Suragan This is me
0000-0003-4789-1982
Kazakhstan

Muhammad Asad Zaighum ^*
0000-0003-2677-1957
Kazakhstan

Early Pub Date

September 14, 2025

Publication Date

September 15, 2025

Submission Date

June 18, 2025

Acceptance Date

September 12, 2025

Published in Issue

Year 2025 Volume: 8 Number: 3

DOI

https://doi.org/10.33205/cma.1716123

IZ

https://izlik.org/JA23DC67DE

Cite

RIS / Bibtex

APA

Nursultanov, E., Suragan, D., & Zaighum, M. A. (2025). On the boundedness of Riesz potential operators: insights from net spaces. Constructive Mathematical Analysis, 8(3), 165-175. https://doi.org/10.33205/cma.1716123

AMA

1.Nursultanov E, Suragan D, Zaighum MA. On the boundedness of Riesz potential operators: insights from net spaces. CMA. 2025;8(3):165-175. doi:10.33205/cma.1716123

Chicago

Nursultanov, Erlan, Durvudkhan Suragan, and Muhammad Asad Zaighum. 2025. “On the Boundedness of Riesz Potential Operators: Insights from Net Spaces”. Constructive Mathematical Analysis 8 (3): 165-75. https://doi.org/10.33205/cma.1716123.

EndNote

Nursultanov E, Suragan D, Zaighum MA (September 1, 2025) On the boundedness of Riesz potential operators: insights from net spaces. Constructive Mathematical Analysis 8 3 165–175.

IEEE

[1]E. Nursultanov, D. Suragan, and M. A. Zaighum, “On the boundedness of Riesz potential operators: insights from net spaces”, CMA, vol. 8, no. 3, pp. 165–175, Sept. 2025, doi: 10.33205/cma.1716123.

ISNAD

Nursultanov, Erlan - Suragan, Durvudkhan - Zaighum, Muhammad Asad. “On the Boundedness of Riesz Potential Operators: Insights from Net Spaces”. Constructive Mathematical Analysis 8/3 (September 1, 2025): 165-175. https://doi.org/10.33205/cma.1716123.

JAMA

1.Nursultanov E, Suragan D, Zaighum MA. On the boundedness of Riesz potential operators: insights from net spaces. CMA. 2025;8:165–175.

MLA

Nursultanov, Erlan, et al. “On the Boundedness of Riesz Potential Operators: Insights from Net Spaces”. Constructive Mathematical Analysis, vol. 8, no. 3, Sept. 2025, pp. 165-7, doi:10.33205/cma.1716123.

Vancouver

1.Erlan Nursultanov, Durvudkhan Suragan, Muhammad Asad Zaighum. On the boundedness of Riesz potential operators: insights from net spaces. CMA. 2025 Sep. 1;8(3):165-7. doi:10.33205/cma.1716123