Boundedness in Lebesgue spaces of Riesz potentials on commutative hypergroups

Mubariz Hajibayov

doi:10.14419/gjma.v3i1.3996

Authors

Mubariz Hajibayov
National Aviation Academy

Received date: December 7, 2014

Accepted date: January 5, 2015

Published date: January 16, 2015

DOI:

https://doi.org/10.14419/gjma.v3i1.3996

Keywords:

Hardy-Littlevood Maximal Function, Hypergroup, Riesz Potential.

Abstract

In the present paper we consider Riesz potentials on commutative hypergroups and prove Â the boundedness of these potentials from \(L^{p} \left( K,\lambda \right) \) Â to \(L^{q} \left( K,\lambda \right) \). We also prove the Â inequality from \(L^{1} \left( K,\lambda \right) \) to weak \(L^{q} \left( K,\lambda \right) \) for Riesz potentials on commutative hypergroups.

References

[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 spacesâ€ 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.