We establish in this paper the Hardy–Littlewood–Sobolev inequalities for the Riesz potentials on Morrey spaces over commutative hypergroups. As a consequence, we are also able to get Olsen-type inequality on the same spaces. Here, the condition of upper Ahlfors n-regular by identity is assumed to obtain the inequalities.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Adams, D.: A note on Riesz potentials. Duke Math. J. 42, 765–778 (1975)
Bloom, W.R., Heyer, H.: Harmonic analysis of probability measures on hypergroups. de Gruyter, Berlin (1995)
Chiarenza, F., Frasca, M.: Morrey spaces and Hardy–Littlewood maximal function. Rend. Mat. Appl. 7, 273–279 (1987)
García-Cuerva, J., Gatto, A.E.: Boundedness properties of fractional integral operators associated to non-doubling measures. Stud. Math. 162, 245–261 (2004)
García-Cuerva, J., Martell, J.M.: Two weight norm inequalities for maximal operators and fractional integrals on non-homogeneous spaces. Indiana Univ. Math. J. 50, 1241–1280 (2001)
Gunawan, H., Eridani: Fractional integrals and generalized Olsen inequalities. Kyungpook Math. J. 49(1), 31–39 (2009)
Gunawan, H.: A note on the generalized fractional integral operators. J. Indones. Math. Soc. (MIHMI) 9(1), 39–43 (2003)
Guliyev, V., Sawano, Y.: Linear and sublinear operators on generalized Morrey spaces with non-doubling measures. Publ. Math. Debr. 83(3), 303–327 (2013)
Guo, Y.: Generalized Olsen inequality and Schrödinger type elliptic equations. Georgian Math. J. (2019). https://doi.org/10.1515/gmj-2019-2059
Hajibayov, M.G.: Boundedness in Lebesgue spaces of Riesz potentials on commutative hypergroups. Glob. J. Math. Anal. 3(1), 18–25 (2015)
Hardy, G.H., Littlewood, J.E.: Some properties of fractional integrals. Math. Z. 27, 565–606 (1927)
Jewett, R.I.: Spaces with an abstract convolution of measure. Adv. Math. 18, 1–101 (1975)
Kurata, K., Nishigaki, S., Sugano, S.: Boundedness of integral operators on generalized Morrey spaces and its application to Schrödinger operators. Proc. Am. Math. Soc. 128, 1125–1134 (2002)
Morrey, C.B.: Functions of several variables and absolute continuity. Duke Math. J. 6, 187–215 (1940)
Nakai, E.: Hardy–Littlewood maximal operator, singular integral operators, and the Riesz potentials on generalized Morrey spaces. Math. Nachr. 166, 95–103 (1994)
Nakai, E.: Generalized fractional integrals on generalized Morrey spaces. Math. Nachr. 287(2–3), 339–351 (2014)
Nakai, E., Sadasue, G.: Martingale Morrey–Campanato spaces and fractional integrals. J. Funct. Spaces Appl. 2012, 29 (2012). (Article ID 673929)
Olsen, P.A.: Fractional integration, Morrey spaces and a Schrödinger equation. Commun. Partial Differ. Equ. 20, 2005–2055 (1995)
Ruzhansky, M., Suragan, D., Yessirkegenov, N.: Hardy–Littlewood, Bessel–Riesz, and fractional integral operators in anisotropic Morrey and Companato spaces. Fract. Calc. Appl. Anal. 21(3), 577–612 (2018)
Sawano, Y., Tanaka, H.: Morrey spaces for non-doubling measures. Acta Math. Sin. (Engl. Ser.) 21(6), 1535–1544 (2005)
Sawano, Y., Shimomura, T.: Sobolev’s inequality for Riesz potentials of functions in generalized Morrey spaces with variable exponent attaining the value \(1\) over non-doubling measure spaces. J. Inequal. Appl. 2013, 1–19 (2013)
Sawano, Y., Sugano, S., Tanaka, H.: Generalized fractional integral operators and fractional maximal operators in the framework of Morrey spaces. Trans. Am. Math. Soc. 263(12), 6481–6503 (2011)
Sihwaningrum, I., Gunawan, H., Nakai, E.: Maximal and fractional integral operators on generalized Morrey spaces over metric measure spaces. Math. Nachr. 291, 1400–1417 (2018)
Sihwaningrum, I., Suryawan, H.P., Gunawan, H.: Fractional integral operators and Olsen inequalities on non-homogeneous spaces. Aust. J. Math. Anal. Appl. 7(1), 14 (2010)
Sihwaningrum, I., Sawano, Y.: Weak and strong type estimates for fractional integral operator on Morrey spaces over metric measure spaces. Eurasian Math. J. 4, 76–81 (2013)
Sobolev, S.L.: On a theorem in functional analysis (Russian). Mat. Sob. 46, 471–497 (1938). (English translation in Amer. Math. Soc. Transl. ser. 2, 34, 39–68 (1963))
We are indebted to the referees for their useful comments on the earlier version of this paper. The first and second authors are supported by Directorate General of Higher Education, Grant/Award Number: P/1721/IJN23T4PN/2019. The third author is supported by P3MI-ITB Program 2020.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
Sihwaningrum, I., Maryani, S. & Gunawan, H. Extension of Hardy–Littlewood–Sobolev Inequalities for Riesz Potentials on Hypergroups. Mediterr. J. Math. 17, 203 (2020). https://doi.org/10.1007/s00009-020-01645-w
- Ahlfors condition
- Riesz potential
- Morrey spaces
- commutative hypergroup
- Olsen inequality
Mathematics Subject Classification