Boundedness in Lebesgue spaces of Riesz potentials on commutative hypergroups
DOI:
https://doi.org/10.14419/gjma.v3i1.3996الكلمات المفتاحية:
Hardy-Littlevood Maximal Function، Hypergroup، Riesz Potential.الملخص
In the present paper we consider Riesz potentials on commutative hypergroups and prove  the boundedness of these potentials from \(L^{p} \left( K,\lambda \right) \)  to \(L^{q} \left( K,\lambda \right) \). We also prove the  inequality from \(L^{1} \left( K,\lambda \right) \) to weak \(L^{q} \left( K,\lambda \right) \) for Riesz potentials on commutative hypergroups.المراجع
[1] W. R. Bloom, H. Heyer, Harmonic analysis of probability measures on hypergroups, de Gruyter Stud. Math., Vol. 20, Walter de Gruyter & Co., Berlin, 1995.
[2] R. R. Coifman and G. Weiss, Analyse harmonique non-commutative sur certains espaces homog`enes.(French) Lecture Notes in Math., Vol.242, Springer-Verlag, Berlin-New York, 1971.
[3] A. D. Gadjiev, â€On generalized potential-type integral operatorsâ€, Functiones et Approximatio, UAM, Vol.25 (1997), pp. 37-44.
[4] A. D. Gadjiev, M. G. Hajibayov, â€Inequalities for B-convolution operatorsâ€, TWMS J. Pure Appl. Math., Vol.1, No.1, (2010), pp.41-52.
[5] J. Garcia-Cuerva, A. E. Gatto, â€Boundedness properties of fractional integral operators associated to non-doubling measuresâ€, Studia Math., Vol.162, (2004), pp.245-261 .
[6] G. Gaudry, S. Giulini, A. Hulanicki, A. M. Mantero, â€Hardy-Littlewood maximal function on some solvable Lie groupsâ€, J. Austral. Math. Soc. Ser. A, Vol.45, No.1, (1988), pp.78-82.
[7] G. Gigante, â€Transference for hypergroupsâ€, Collect. Math., Vol.52, No.2, (2001), pp.127-155.
[8] V. S.Guliyev, â€On maximal function and fractional integral, associated with the Bessel differential operatorâ€, Math. Inequal. Appl., Vol.6, No.2, (2003), pp.317-330.
[9] V. S. Guliyev, M.N. Omarova, â€On fractional maximal function and fractional integral on the Laguerre hypergroupâ€, J. Math. Anal. Appl., Vol.340, No.2, (2008), pp.1058-1068.
[10] V. S. Guliyev, Y. Y. Mammadov, â€On fractional maximal function and fractional integrals associated with the Dunkl operator on the real lineâ€, J. Math. Anal. Appl., Vol.353, (2009), pp.449-459.
[11] M.G.Hajibayov, â€(Lp;Lq) properties of the potential-type integrals associated to non-doubling measuresâ€, Sarajevo J. Math., Vol.2, No.15, (2006), pp.173-180.
[12] M. G. Hajibayov, S. G. Samko, â€Weighted estimates of generalized potentials in variable exponent Lebesgue spaces on homogeneous spaces†Operator Theory: Adv. and Appl., Vol.210, (2010), pp.107-122.
[13] M. G. Hajibayov, S. G. Samko, â€Generalized potentials in variable exponent Lebesgue spaces on homogeneous spacesâ€, Math. Nachr., Vol.284, No.1, (2011), pp.53-66.
[14] M.G.Hajibayov, â€Boundedness of the Dunkl convolution operatorsâ€, An. Univ. Vest Timis,. Ser. Mat.-Inform., Vol.49, No.1, (2011), pp.49-67.
[15] M.G.Hajibayov, â€Inequalities for convolutions of functions on commutative hypergroupsâ€, Azerb. J. Math., Vol.4, No.1, (2014), pp.92-107. arXiv preprint arXiv:1307.4948, 2013.
[16] L. Hedberg, â€On certain convolution inequalitiesâ€, Proc. Amer. Math. Soc., Vol.36, (1972), pp.505-510.
[17] R. L. Jewett, â€Spaces with an abstract convolution of measuresâ€, Adv. in Math., Vol.18, No.1, (1975), pp.1-101.
[18] V. M. Kokilashvili, A. Kufner, â€Fractional inteqrals on spaces of homogeneous typeâ€, Comment. Math. Univ. Carolinae, Vol.30, No.3, (1989), pp.511-523.
[19] V. Kokilashvili, A. Meskhi, â€Fractional integrals on measure spacesâ€, Frac. Calc. Appl. Anal., Vol.4, No.1, (2001), pp.1-24.
[20] M. Lashkarizadeh Bami, â€The semisimplicity of L1(K;w) of a weighted commutative hypergroup Kâ€, Acta Math. Sinica, Vol.24, No.4, (2008), pp.607-610.
[21] E. Nakai, â€On generalized fractional integralsâ€, Taiwanese Math. J., Vol.5, No.3, (2001), pp.587-602.
[22] E. Nakai, H. Sumitomo, â€On generalized Riesz potentials and spaces of some smooth functionsâ€, Sci. Math. Japonicae, Vol.54, No.3, (2001), pp.463-472.
[23] R. Spector, â€Measures invariantes sur les hypergroupes(French)â€, Trans. Amer. Math. Soc., Vol.239, (1978), pp.147-165.
[24] E. Stein, Singular integrals and diferentiability properties of functions, Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J., 1970.
التنزيلات
منشور
إصدار
القسم
الرخصة
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal''s published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
