The completeness role of the function ϕ in generating the Riesz potential operator
##plugins.themes.bootstrap3.article.main##
Abstract
The Riesz potential operator is a central tool in harmonic analysis and the theory of partial differential equations, commonly defined via convolution with a singular Kernel. In many modern frameworks, function space are generated by a mappings involving such operators. In this paper, we explore the dual role of the generating function- in: (i). Defining the Riesz function space and (ii). Ensuring its completeness. We introduce a Riesz function space whose norm is induced growth function- (a Young function). We establish, through several examples and proofs, that under suitable conditions (specifically, the condition on ), the space is complete. Furthemore, we illustrate discrete analogues and applications to Orlicz space, thereby underscoring the fundamental importance of in both the construction and Banach space structure of these function spaces.
##plugins.themes.bootstrap3.article.details##
How to Cite
Vinsensia, D., Utami, Y., & Addini, P. F. (2025). The completeness role of the function ϕ in generating the Riesz potential operator. International Journal of Basic and Applied Science, 13(4), 171–178. https://doi.org/10.35335/ijobas.v13i4.637
References
D. R. Adam and L. I. Hedberg, Function Spaces and Potential Theory. Derlin: Springer Berlin Heidelberg.
L. Grafakos, Classical Fourier Analysis, Third. London: Springer, 2014.
E. M. Stein, Singular Integrals and Differentiability Properties of Function, Third. New Jersey: Princeton University Press, 1970.
E. Burtseva and L. Maligranda, “A new result on boundedness of the Riesz potential in central Morrey–Orlicz spaces,” Positivity, vol. 27, no. 5, pp. 1–27, 2023, doi: 10.1007/s11117-023-01013-4.
K. Dorak, F. Deringoz, and V. S. Guliyev, “Boundedness of the maximal operator and the Riesz potential on Musielak-Orlicz-Morrey spaces,” vol. XX, no. x, pp. 1–12, 2025.
M. A. Krasnosel’skii and B. Rutickii, YA, “Convex Functions and Orlicz Spaces,” Cambridge Univ. Press, vol. 47, no. 361, pp. 266–267, 1961.
P. Harjulehto and P. Hästö, Sobolev Spaces, vol. 2236. 2019. doi: 10.1007/978-3-030-15100-3_6.
X. Hao, B. Li, and S. Yang, “Estimates of discrete Riesz potentials on discrete weighted Lebesgue spaces,” Ann. Funct. Anal., vol. 15, no. 3, pp. 1–21, 2024, doi: 10.1007/s43034-024-00357-6.
C. Bennett and R. C. Sharpley, Interpolation of operators, vol. 129. Academic press, 1988.
A. Cruz-Uribe, D. V., & Fiorenza, Variable Lebesgue Spaces. New York: Springer Heidelberg New York Dordrecht London, 2013. doi: doi:10.1007/978-3-0348-0548-3.
L. Diening, P. Harjulehto, P. Hästö, and M. Ruzicka, Lebesgue and Sobolev Spaces with Variable Exponents. Springer, 2011.
K. P. Ho, “Operators on Herz–Morrey Spaces With Variable Exponents,” Math. Inequalities Appl., vol. 26, no. 4, pp. 861–886, 2023, doi: 10.7153/mia-2023-26-53.
H. Kalita and B. Hazarika, “Modular convergence in H-Orlicz spaces of Banach valued functions,” Rend. del Circ. Mat. di Palermo, vol. 72, no. 8, pp. 3905–3916, 2023, doi: 10.1007/s12215-023-00871-x.
T. Zhou and Y. Lei, “On discrete reversed Hardy-Littlewood-Sobolev inequalities,” Can. Math. Bull., pp. 1–12, Mar. 2025, doi: 10.4153/S0008439525000293.
E. H. Lieb and M. Loss, Analysis, vol. 14. American Mathematical Soc., 2001.
J. Musielak, O. Spaces, and M. Spaces, Orlicz Spaces and Modular Spaces. Springer Basel, 1983.
P. Wojtaszczyk, Banach Spaces for Analysts. Cambridge: Cambridge University Press, 1991. doi: 10.1017/CBO9780511608735.
Z. D. R. M.M. Rao, Theory of Orlicz Spaces. New york: CRC Press, 2002. doi: https://doi.org/10.1201/9780203910863.
K. Yosida, Functional Analysis, Sixth. Springer Berlin Heidelberg, 1995. doi: 10.1007/978-3-642-61859-8.
D. Yang and S. Yang, “Real-variable characterizations of Orlicz-Hardy spaces on strongly Lipschitz domains of Rn,” Rev. Mat. Iberoam., vol. 29, no. 1, 2013, doi: 10.4171/rmi/719.
X. Yan, Z. He, D. Yang, and W. Yuan, “Hardy spaces associated with ball quasi-Banach function spaces on spaces of homogeneous type: Characterizations of maximal functions, decompositions, and dual spaces,” Math. Nachrichten, vol. 296, no. 7, 2023, doi: 10.1002/mana.202100432.
Z. D. R. M.M. Rao, Applications Of Orlicz Spaces. New Jersey: Marcel Decker Inc, 2002.
E. Sawyer, “Boundedness of classical operators on classical Lorentz spaces,” Stud. Math., vol. 96, no. 2, pp. 145–158, 1990.
N. Uthirasamy, K. Tamilvanan, H. K. Nashine, and R. George, “Solution and Stability of Quartic Functional Equations in Modular Spaces by Using Fatou Property,” J. Funct. Spaces, vol. 2022, 2022, doi: 10.1155/2022/5965628.
G. Di Fratta and A. Fiorenza, “A direct approach to the duality of grand and small Lebesgue spaces,” Nonlinear Anal. Theory, Methods Appl., vol. 70, no. 7, 2009, doi: 10.1016/j.na.2008.03.044.
V. Kokilashvili and A. Meskhi, “Maximal and singular integral operators in weighted grand variable exponent Lebesgue spaces,” Ann. Funct. Anal., vol. 12, no. 3, 2021, doi: 10.1007/s43034-021-00135-8.
G. B. Folland, “Real Analysis: Modern Techniques & Their Applications,” Top, vol. 9, no. 1. 2001.
M. Ledoux and M. Talagrand, Probability in Banach Spaces: Isoperimetry and Processes. 1991.
L. Grafakos, Classical Fourier Analysis, Third. London: Springer, 2014.
E. M. Stein, Singular Integrals and Differentiability Properties of Function, Third. New Jersey: Princeton University Press, 1970.
E. Burtseva and L. Maligranda, “A new result on boundedness of the Riesz potential in central Morrey–Orlicz spaces,” Positivity, vol. 27, no. 5, pp. 1–27, 2023, doi: 10.1007/s11117-023-01013-4.
K. Dorak, F. Deringoz, and V. S. Guliyev, “Boundedness of the maximal operator and the Riesz potential on Musielak-Orlicz-Morrey spaces,” vol. XX, no. x, pp. 1–12, 2025.
M. A. Krasnosel’skii and B. Rutickii, YA, “Convex Functions and Orlicz Spaces,” Cambridge Univ. Press, vol. 47, no. 361, pp. 266–267, 1961.
P. Harjulehto and P. Hästö, Sobolev Spaces, vol. 2236. 2019. doi: 10.1007/978-3-030-15100-3_6.
X. Hao, B. Li, and S. Yang, “Estimates of discrete Riesz potentials on discrete weighted Lebesgue spaces,” Ann. Funct. Anal., vol. 15, no. 3, pp. 1–21, 2024, doi: 10.1007/s43034-024-00357-6.
C. Bennett and R. C. Sharpley, Interpolation of operators, vol. 129. Academic press, 1988.
A. Cruz-Uribe, D. V., & Fiorenza, Variable Lebesgue Spaces. New York: Springer Heidelberg New York Dordrecht London, 2013. doi: doi:10.1007/978-3-0348-0548-3.
L. Diening, P. Harjulehto, P. Hästö, and M. Ruzicka, Lebesgue and Sobolev Spaces with Variable Exponents. Springer, 2011.
K. P. Ho, “Operators on Herz–Morrey Spaces With Variable Exponents,” Math. Inequalities Appl., vol. 26, no. 4, pp. 861–886, 2023, doi: 10.7153/mia-2023-26-53.
H. Kalita and B. Hazarika, “Modular convergence in H-Orlicz spaces of Banach valued functions,” Rend. del Circ. Mat. di Palermo, vol. 72, no. 8, pp. 3905–3916, 2023, doi: 10.1007/s12215-023-00871-x.
T. Zhou and Y. Lei, “On discrete reversed Hardy-Littlewood-Sobolev inequalities,” Can. Math. Bull., pp. 1–12, Mar. 2025, doi: 10.4153/S0008439525000293.
E. H. Lieb and M. Loss, Analysis, vol. 14. American Mathematical Soc., 2001.
J. Musielak, O. Spaces, and M. Spaces, Orlicz Spaces and Modular Spaces. Springer Basel, 1983.
P. Wojtaszczyk, Banach Spaces for Analysts. Cambridge: Cambridge University Press, 1991. doi: 10.1017/CBO9780511608735.
Z. D. R. M.M. Rao, Theory of Orlicz Spaces. New york: CRC Press, 2002. doi: https://doi.org/10.1201/9780203910863.
K. Yosida, Functional Analysis, Sixth. Springer Berlin Heidelberg, 1995. doi: 10.1007/978-3-642-61859-8.
D. Yang and S. Yang, “Real-variable characterizations of Orlicz-Hardy spaces on strongly Lipschitz domains of Rn,” Rev. Mat. Iberoam., vol. 29, no. 1, 2013, doi: 10.4171/rmi/719.
X. Yan, Z. He, D. Yang, and W. Yuan, “Hardy spaces associated with ball quasi-Banach function spaces on spaces of homogeneous type: Characterizations of maximal functions, decompositions, and dual spaces,” Math. Nachrichten, vol. 296, no. 7, 2023, doi: 10.1002/mana.202100432.
Z. D. R. M.M. Rao, Applications Of Orlicz Spaces. New Jersey: Marcel Decker Inc, 2002.
E. Sawyer, “Boundedness of classical operators on classical Lorentz spaces,” Stud. Math., vol. 96, no. 2, pp. 145–158, 1990.
N. Uthirasamy, K. Tamilvanan, H. K. Nashine, and R. George, “Solution and Stability of Quartic Functional Equations in Modular Spaces by Using Fatou Property,” J. Funct. Spaces, vol. 2022, 2022, doi: 10.1155/2022/5965628.
G. Di Fratta and A. Fiorenza, “A direct approach to the duality of grand and small Lebesgue spaces,” Nonlinear Anal. Theory, Methods Appl., vol. 70, no. 7, 2009, doi: 10.1016/j.na.2008.03.044.
V. Kokilashvili and A. Meskhi, “Maximal and singular integral operators in weighted grand variable exponent Lebesgue spaces,” Ann. Funct. Anal., vol. 12, no. 3, 2021, doi: 10.1007/s43034-021-00135-8.
G. B. Folland, “Real Analysis: Modern Techniques & Their Applications,” Top, vol. 9, no. 1. 2001.
M. Ledoux and M. Talagrand, Probability in Banach Spaces: Isoperimetry and Processes. 1991.
Copyright and Licensing
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.