Home Weak and strong type inequalities criteria for fractional maximal functions and fractional integrals associated with Gegenbauer differential operator
Article
Licensed
Unlicensed Requires Authentication

Weak and strong type inequalities criteria for fractional maximal functions and fractional integrals associated with Gegenbauer differential operator

  • Vakhtang Kokilashvili and Elman J. Ibrahimov EMAIL logo
Published/Copyright: June 6, 2023
Become an author with De Gruyter Brill

Abstract

This paper deals with the introduction and study of a new, Orlicz type space associated with the Gegenbauer differential operator and briefly called here a G-Orlicz space. We prove that such spaces are Banach function spaces. The necessary and sufficient conditions are established which ensure the validity of weak and strong type inequalities for fractional maximal functions and fractional integrals of G-Orlicz spaces.

References

[1] C. Bennett and R. Sharpley, Interpolation of Operators, Pure Appl. Math. 129, Academic Press, Boston, 1988. Search in Google Scholar

[2] A. Cianchi, A sharp embedding theorem for Orlicz–Sobolev spaces, Indiana Univ. Math. J. 45 (1996), no. 1, 39–65. 10.1512/iumj.1996.45.1958Search in Google Scholar

[3] A. Cianchi, Strong and weak type inequalities for some classical operators in Orlicz spaces, J. Lond. Math. Soc. (2) 60 (1999), no. 1, 187–202. 10.1112/S0024610799007711Search in Google Scholar

[4] D. E. Edmunds and W. D. Evans, Hardy Operators, Function Spaces and Embeddings, Springer Monogr. Math., Springer, Berlin, 2004. 10.1007/978-3-662-07731-3Search in Google Scholar

[5] V. S. Guliev and E. D. Ibragimov, Conditions for the L p , λ -boundedness of the Riesz potential generated by the Gegenbauer differential operator, Mat. Zametki 105 (2019), no. 5, 685–695; translation in Math. Notes 105 (2019), no. 5-6, 674–683. Search in Google Scholar

[6] V. S. Guliyev and E. J. Ibrahimov, Necessary and sufficient condition for the boundedness of the Gegenbauer–Riesz potential on Morrey spaces, Georgian Math. J. 25 (2018), no. 2, 235–248. 10.1515/gmj-2018-0022Search in Google Scholar

[7] V. S. Guliyev, E. J. Ibrahimov, S. E. Ekincioglu and S. A. Jafarova, O’Neil inequality for convolutions associated with Gegenbauer differential operator and some applications, J. Math. Study 53 (2020), no. 1, 90–124. 10.4208/jms.v53n1.20.05Search in Google Scholar

[8] P. Harjulehto and P. Hästö, Orlicz Spaces and Generalized Orlicz Spaces, Lecture Notes in Math. 2236, Springer, Cham, 2019. 10.1007/978-3-030-15100-3Search in Google Scholar

[9] E. J. Ibrahimov, On Gegenbauer transformation on the half-line, Georgian Math. J. 18 (2011), no. 3, 497–515. 10.1515/gmj.2011.0024Search in Google Scholar

[10] E. J. Ibrahimov and A. Akbulut, The Hardy–Littlewood–Sobolev theorem for Riesz potential generated by Gegenbauer operator, Trans. A. Razmadze Math. Inst. 170 (2016), no. 2, 166–199. 10.1016/j.trmi.2016.05.004Search in Google Scholar

[11] E. J. Ibrahimov, G. A. Dadashova and S. A. Jafarova, Some weighted inequalities for Gegenbauer fractional integrals, Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. 41 (2021), no. 4, 83–99. Search in Google Scholar

[12] E. J. Ibrahimov, V. S. Guliyev and S. A. Jafarova, Weighted boundedness of the fractional maximal operator and Riesz potential generated by Gegenbauer differential operator, Trans. A. Razmadze Math. Inst. 173 (2019), no. 3, 45–78. Search in Google Scholar

[13] H. Kita, On maximal functions in Orlicz spaces, Proc. Amer. Math. Soc. 124 (1996), no. 10, 3019–3025. 10.1090/S0002-9939-96-03807-5Search in Google Scholar

[14] H. Kita, On Hardy–Littlewood maximal functions in Orlicz spaces, Math. Nachr. 183 (1997), 135–155. 10.1002/mana.19971830109Search in Google Scholar

[15] V. Kokilashvili and M. Krbec, Weighted Inequalities in Lorentz and Orlicz Spaces, World Scientific, River Edge, 1991. 10.1142/1367Search in Google Scholar

[16] W. A. J. Luxemburg, Banach Function Spaces, Thesis, Technische Hogeschool te Delft, Delft, 1955. Search in Google Scholar

[17] L. Maligranda, Orlicz Spaces and Interpolation, Semin. Mat. 5, Universidade Estadual de Campinas, Campinas, 1989. Search in Google Scholar

[18] E. Nakai, On generalized fractional integrals in the Orlicz spaces on spaces of homogeneous type, Sci. Math. Jpn. 54 (2001), no. 3, 473–487. Search in Google Scholar

[19] H. Nakano, Modulared Semi-Ordered Linear Spaces, Maruzen, Tokyo, 1950. Search in Google Scholar

[20] R. O’Neil, Fractional integration in Orlicz spaces. I, Trans. Amer. Math. Soc. 115 (1965), 300–328. 10.1090/S0002-9947-1965-0194881-0Search in Google Scholar

[21] W. Orlicz, Über gewisse Klassen von Modularräumen, Bull. Int. Acad. Polon. Sci. A 1932 (1962), no. 8–9, 207–220. Search in Google Scholar

[22] W. Orlicz, Über Räume ( L M ) , Bull. Int. Acad. Polon. Sci. A 1936 (1936), 93–107. Search in Google Scholar

[23] M. M. Rao and Z. D. Ren, Applications of Orlicz Spaces, Monog. Textb. Pure Appl. Math. 250, Marcel Dekker, New York, 2002. 10.1201/9780203910863Search in Google Scholar

[24] A. Torchinsky, Interpolation of operations and Orlicz classes, Studia Math. 59 (1976/77), no. 2, 177–207. 10.4064/sm-59-2-177-207Search in Google Scholar

[25] N. S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967), 473–483. 10.1512/iumj.1968.17.17028Search in Google Scholar

Received: 2022-04-28
Revised: 2022-06-15
Accepted: 2022-11-21
Published Online: 2023-06-06
Published in Print: 2023-10-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 25.10.2025 from https://www.degruyterbrill.com/document/doi/10.1515/gmj-2023-2031/pdf
Scroll to top button