Abstract
This paper deals with the hyperbolic Riesz B-potential, which is the negative real power of an operator B γ1 − ∑ n i = 2 B γi , where \( \begin{array}{cc}\hfill {B}_{\upgamma i}={\partial}^2/\partial {x}_i^2+\left({\upgamma}_i/{x}_i\right)\partial /\partial {x}_i,\hfill & \hfill i=1,\dots, n,\hfill \end{array} \) is a singular Bessel differential operator. We prove the boundedness of the hyperbolic Riesz B-potential in proper spaces.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J. Bergh and J. Löfström, Interpolation Spaces: An Introduction, Springer-Verlag, Berlin, Heidelberg, New York, 1976.
O.V. Besov, V.P. Il’in, and S.M. Nikol’skii, Integral Representation of Functions and Imbedding Theorems, Vol. I, V.H. Winston, Washington, 1978.
I.A. Kipriyanov, Singular Elliptic Boundary Value Problems, Nauka, Moscow, 1997 (in Russian).
B.M. Levitan, Expansion in Fourier series and integrals with Bessel functions, Usp. Mat. Nauk, 6(2(42)):102–143, 1951 (in Russian).
L.N. Lyakhov, Inversion of Riesz B-potentials, Dokl. Akad. Nauk SSSR, 321(3):466–469, 1991 (in Russian).
L.N. Lyakhov, Spaces of Riesz B-potentials, Dokl. Akad. Nauk SSSR, 334(3):278–280, 1994 (in Russian).
L.N. Lyakhov, Symbol of the integral operator of Riesz B-potential with single characteristic, Dokl. Akad. Nauk, 351(2):164–168, 1996 (in Russian).
L.N. Lyakhov and E.L. Shishkina, Generalized Riesz B-potentials of the mixed type, Dokl. Math., 73(1):42–45, 2006.
L.N. Lyakhov and E.L. Shishkina, Inversion of general Riesz B-potentials, in M.V. Dubatovskaya and S.V. Rogosin (Eds.), The 7th InternationalWorkshop on AnalyticalMethods of Analysis and Differential Equations, Vol. 10,Minsk, Belarus, September 10–15, 2012, Cambridge Scientific Publishers, Cottenham, 2013, pp. 115–126.
L.N. Lyakhov and E.L. Shishkina, Marcinkiewicz interpolation theorem for weighted Lebesgue classes, Vestn. Voronezh. Gos. Univ., Ser. Fiz. Mat., 10(1):159–169, 2015 (in Russian).
V.A. Nogin and E.V. Sukhinin, Inversion and characterization of hyperbolic potentials in L p -spaces, Dokl. Akad. Nauk, Ross. Akad. Nauk, 329(5):550–552, 1993 (in Russian).
M. Riesz, Intégrale de Riemann–Liouville et solution invariantive du problème de Cauchy pour l’équation de sondes, in Comptes Rendus du Congrès International des Mathematiciens, Oslo 1936, Vol. II, A.W. Brøggers Boktrykkeri A/S, Oslo, 1937, pp. 44–45.
M. Riesz, L’intégrale de Riemann–Liouville et le probléme de Cauchy, Acta Math., 81:1–223, 1949.
S.G. Samko, A.A. Kilbas, and O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon & Breach, Yverdon, 1993.
M.Z. Sarikaya, H. Yildirim, and Ö. Akin, On generalized Riesz type potential with Lorentz distance, Lobachevskii J. Math., 29(1):32–39, 2008.
E.L. Shishkina, Boundedness of potential operators with hyperbolic distance, in Abstracts of Reports of the 8th International Workshop AMADE-2015, Minsk, Belarus, September 14–19, 2015, Institute of Mathematics, NASB, 2015, p. 90.
Ya.I. Zhitomirskii, Cauchy’s problem for systems of linear partial differential equations with differential operators of Bessel type, Mat. Sb., Nov. Ser., 36(78)(2):299–310, 1955.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Shishkina, E.L. On the boundedness of hyperbolic Riesz B-potential. Lith Math J 56, 540–551 (2016). https://doi.org/10.1007/s10986-016-9335-y
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10986-016-9335-y