Boundedness in Lebesgue spaces of Riesz potentials on commutative hypergroups

  • Abstract
  • Keywords
  • References
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  • Abstract

    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.

  • Keywords

    Hardy-Littlevood Maximal Function; Hypergroup; Riesz Potential.

  • 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, ”On generalized potential-type integral operators”, Functiones et Approximatio, UAM, Vol.25 (1997), pp. 37-44.

      [4] A. D. Gadjiev, M. G. Hajibayov, ”Inequalities for B-convolution operators”, TWMS J. Pure Appl. Math., Vol.1, No.1, (2010), pp.41-52.

      [5] J. Garcia-Cuerva, A. E. Gatto, ”Boundedness properties of fractional integral operators associated to non-doubling measures”, Studia Math., Vol.162, (2004), pp.245-261 .

      [6] G. Gaudry, S. Giulini, A. Hulanicki, A. M. Mantero, ”Hardy-Littlewood maximal function on some solvable Lie groups”, J. Austral. Math. Soc. Ser. A, Vol.45, No.1, (1988), pp.78-82.

      [7] G. Gigante, ”Transference for hypergroups”, Collect. Math., Vol.52, No.2, (2001), pp.127-155.

      [8] V. S.Guliyev, ”On maximal function and fractional integral, associated with the Bessel differential operator”, Math. Inequal. Appl., Vol.6, No.2, (2003), pp.317-330.

      [9] V. S. Guliyev, M.N. Omarova, ”On fractional maximal function and fractional integral on the Laguerre hypergroup”, J. Math. Anal. Appl., Vol.340, No.2, (2008), pp.1058-1068.

      [10] V. S. Guliyev, Y. Y. Mammadov, ”On fractional maximal function and fractional integrals associated with the Dunkl operator on the real line”, J. Math. Anal. Appl., Vol.353, (2009), pp.449-459.

      [11] M.G.Hajibayov, ”(Lp;Lq) properties of the potential-type integrals associated to non-doubling measures”, Sarajevo J. Math., Vol.2, No.15, (2006), pp.173-180.

      [12] M. G. Hajibayov, S. G. Samko, ”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, ”Generalized potentials in variable exponent Lebesgue spaces on homogeneous spaces”, Math. Nachr., Vol.284, No.1, (2011), pp.53-66.

      [14] M.G.Hajibayov, ”Boundedness of the Dunkl convolution operators”, An. Univ. Vest Timis,. Ser. Mat.-Inform., Vol.49, No.1, (2011), pp.49-67.

      [15] M.G.Hajibayov, ”Inequalities for convolutions of functions on commutative hypergroups”, Azerb. J. Math., Vol.4, No.1, (2014), pp.92-107. arXiv preprint arXiv:1307.4948, 2013.

      [16] L. Hedberg, ”On certain convolution inequalities”, Proc. Amer. Math. Soc., Vol.36, (1972), pp.505-510.

      [17] R. L. Jewett, ”Spaces with an abstract convolution of measures”, Adv. in Math., Vol.18, No.1, (1975), pp.1-101.

      [18] V. M. Kokilashvili, A. Kufner, ”Fractional inteqrals on spaces of homogeneous type”, Comment. Math. Univ. Carolinae, Vol.30, No.3, (1989), pp.511-523.

      [19] V. Kokilashvili, A. Meskhi, ”Fractional integrals on measure spaces”, Frac. Calc. Appl. Anal., Vol.4, No.1, (2001), pp.1-24.

      [20] M. Lashkarizadeh Bami, ”The semisimplicity of L1(K;w) of a weighted commutative hypergroup K”, Acta Math. Sinica, Vol.24, No.4, (2008), pp.607-610.

      [21] E. Nakai, ”On generalized fractional integrals”, Taiwanese Math. J., Vol.5, No.3, (2001), pp.587-602.

      [22] E. Nakai, H. Sumitomo, ”On generalized Riesz potentials and spaces of some smooth functions”, Sci. Math. Japonicae, Vol.54, No.3, (2001), pp.463-472.

      [23] R. Spector, ”Measures invariantes sur les hypergroupes(French)”, 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.




Article ID: 3996
DOI: 10.14419/gjma.v3i1.3996

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