INVERSION OF THE MIXED RIESZ
HYPERBOLIC B-POTENTIALS
Elina L. Shishkina
Universitetskaya pl., 1, room 223
Voronezh - 394006, RUSSIA

Abstract

In this article a theory of fractional powers of a singular hyperbolic operator on arbitrary spaces is discussed. We consider the problem of inversion of the mixed hyperbolic Riesz B-potential operator on weighted Lebesgue spaces. We apply here the method of approximative inverses.

Citation details of the article



Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 30
Issue: 6
Year: 2017

DOI: 10.12732/ijam.v30i6.3

Download Section



Download the full text of article from here.

You will need Adobe Acrobat reader. For more information and free download of the reader, please follow this link.

References

  1. [1] M. Riesz, Intégrale de Riemann-Liouville et solution invariantive du probléme de Cauchy pour l’équation de sondes, Comptes Rendus du Congres International des Mathematiciens, 2 (1936), 44-45.
  2. [2] M. Riesz, L’integrale de Riemann-Liouville et le probleme de Cauchy, Acta Mathematica, 81 (1949), 1-223.
  3. [3] L. Schwartz, Théorie des distributions, Hermann, 1966 (1st Ed. 1950-1951).
  4. [4] E.M. Stein, Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton, 1970.
  5. [5] S. Helgason, Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions, American Mathematical Soc., 1984.
  6. [6] S.G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach Science, Amsterdam, 1987.
  7. [7] B. Rubin, Fractional Integrals and Potentials, Chapman and Hall/CRC, 1996.
  8. [8] A.A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematical Studies, Vol. 204, Elsevier (North-Holland) Science Publishers, Amsterdam, London and New York, 2006.
  9. [9] M.D. Ortigueira, Fractional Calculus for Scientists and Engineers, Springer Netherlands, Series: Lecture Notes in Electrical Engineering, 84, 2011.
  10. [10] D. Baleanu, K. Diethelm, E. Scalas, J.J. Trujillo, Fractional Calculus: Models and Numerical Methods, World Scientific, 2012.
  11. [11] S.R. Umarov, Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols, Springer International Publishing, Series: Developments in Mathematics, 41, 2015.
  12. [12] V.S. Guliev, Sobolev theorems for B-Riesz potentials, Dokl. RAN, 358, No 4 (1998), 450–451 (In Russian). INVERSION OF THE MIXED RIESZ... 497
  13. [13] V.S. Guliev, Some properties of the anisotropic Riesz-Bessel potential, Analysis Mathematica, 26, No 2 (2000), 20 pp.
  14. [14] D.D. Gasanov, V.S. Guliev, A.X. Narimanov, Certain inequality for the anizotropic B-maximal functions and the anisotropic B-Riesz potentials, Trydi mst. Math. and Mech. Akad. Nauk Azerb. Rep., 4 (1996), 18-24.
  15. [15] A.D. Gadjiev, V.S. Guliyev, A. Serbetci, E.V. Guliyev, The Stein-Weiss type inequalities for the B-Riesz potentials, J. Math. Inequal., 5, No 1 (2011), 87-106.
  16. [16] E.V. Guliyev, Weighted inequality for fractional maximal functions and fractional integrals, associated with the Laplace-Bessel differential operator, Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci., 26, No 1 (2006), 71-80.
  17. [17] V.S. Guliyev and J.J. Hasanov, Sobolev-Morrey type inequality for Riesz potentials, associated with the Laplace-Bessel differential operator, Fract. Calc. Appl. Anal., 9, No 1 (2006), 17-32.
  18. [18] V.S. Guliyev, M. Assal, On maximal function on the Laguerre hypergroup, Fract. Calc. Appl. Anal., 9, No 3 (2006), 1-12.
  19. [19] V.S. Guliev, On maximal function and fractional integral, associated with the Bessel differential operator, Math. Inequal. Appl., 6, No 2 (2003), 317-330.
  20. [20] L.N. Lyakhov, Inversion of the B-Riesz potentials, Dokl. Akad. Nauk SSSR, 321, No 3 (1991), 466-469 (In Russian).
  21. [21] L.N. Lyakhov, Spaces of Riezs B-potentials, Dokl. Akad. Nauk SSSR, 334, No 3 (1994), 278-280 (In Russian).
  22. [22] L.N. Lyakhov, Symbol of the integral operator of Riezs B-potential with single characteristic, Dokl. Akad. Nauk, 351, No 2 (1996), 164-168 (In Russian).
  23. [23] L.N. Lyakhov, E.L. Shishkina, Generalized Riesz B-potentials of the mixed type, Dokl. Akad. Nauk, 73, No 1 (2006), 42-45.
  24. [24] L.N. Lyakhov, E.L. Shishkina, General B-hypersingular integrals with homogeneous characteristic, Dokl. Akad. Nauk, 75, No 1 (2007), 39-43. 498 E.L. Shishkina
  25. [25] L.N. Lyakhov, E.L. Shishkina, Inversion of general Riesz B-potentials with homogeneous characteristic in weight classes of functions, Doklady Akademii Nauk, 426, No 4 (2009), 443-447 (in Russian).
  26. [26] E.L. Shishkina, Inversion of integral of B-potential type with density from Φγ , Journal of Mathematical Sciences, 160, No 1 (2009), 95-102.
  27. [27] V. Kiryakova, Generalized Fractional Calculus and Applications, Pitman Res. Notes Math. 301, Longman Scientific & Technical, Harlow, Co-publ. John Wiley, New York, 1994.
  28. [28] V.A. Nogin and E.V. Sukhinin, Inversion and characterization of hyperbolic potentials in Lp -spaces, Deponierted in VINITI, 2512, No 92 (1992), 1-50 (in Russian).
  29. [29] V.A. Nogin and E.V. Sukhinin, Inversion and characterization of hyperbolic potentials in Lp -spaces, Dokl. Acad. Nauk, 329, No 5 (1993), 550-552 (in Russian).
  30. [30] M.Z. Sarikaya, H. Yildirim, Ö. Akin, On generalized Riesz type potential with Lorentz distance, Lobachevskii Journal of Mathematics, 28 (2008), 24-31.
  31. [31] E.L. Shishkina, On the boundedness of hyperbolic Riesz B-potential, Lithuanian Mathematical Journal, 56, No 4 (2016), 540-551.
  32. [32] E.L. Shishkina, About properties of the averaging kernel in the Lebesgue weighted class, Scientific bulletins of the Belgorod State University. Series: Mathematics. Physics, 42, No 6 (227) (2016), 12-19 (in Russian).
  33. [33] E.L. Shishkina, Integral representation of the kernel of the operator approximating the inverse operator for the hyperbolic Riesz B-potential, Bulletin of Tambov University. Series: Natural and Technical Sciences, 21, No 2 (2016), 448-454 (in Russian).
  34. [34] B.M. Levitan, Expansion in Fourier series and integrals with Bessel functions, Uspekhi Mat. Nauk, 6, No 2 (42) (1951), 102-143 (in Russian).
  35. [35] S.M. Sitnik, Buschman-Erdélyi transmutations: classification, main properties and applications, Belgorod State University Scientific bulletin. Mathematics, Physics, 11, No 208:39 (2015), 60-76. INVERSION OF THE MIXED RIESZ... 499
  36. [36] S.M. Sitnik, On solution to the problem of unitary generalization to the Sonine-Poisson transmutations, Belgorod State University Scientific bulletin. Mathematics, Physics, 5, No 76:18 (2010), 135-153.
  37. [37] S.M. Sitnik, Buschman-Erdélyi transmutations, classification and applications, In: Analytic Methods Of Analysis and Differential Equations: AMADE 2012 (Edited by M.V. Dubatovskaya, S.V. Rogosin), Cambridge Scientific Publishers, Cottenham, Cambridge (2013), 171-201.
  38. [38] S.M. Sitnik, A survey of Buschman–Erdélyi transmutations, Chelyabinskiy Fiziko-Matematicheskiy Zhurnal, 1, No 4 (2016), 63-93.
  39. [39] Ya.I. Zhitomirskii, Cauchy’s problem for systems of linear partial differential equations with differential operators of Bessel type, Mat. Sb. (N. S.), 36, No 78:2 (1955), 299-310.
  40. [40] I.A. Kipriyanov, Singular Elliptic Boundary Value Problems, Nauka, Moscow, 1997.
  41. [41] A.B. Muravnik, Functional differential parabolic equations: integral transformations and qualitative properties of solutions of the Cauchy problem, Journal of Mathematical Sciences, 216, No 3 (2016), 345-496.
  42. [42] I.A. Kipriyanov, L.A. Ivanov, Riezs potentials on the Lorentz spaces, Mat. Sb., 130, No 172:4(8) (1986), 465-474.
  43. [43] S.G. Samko, S.M. Umarkhadzhiev, Description of a space of Riesz potentials in terms of higher derivatives, Soviet Math. (Izv. VUZ), 24, No 11 (1980), 95-98.
  44. [44] S.M. Umarkhadzhiev, Boundedness of the Riesz potential operator in weighted grand Lebesgue spaces, Vladikavkaz. Mat. Zh., 16, No 2 (2014), 62-68.
  45. [45] A.N. Karapetyants, V.A. Nogin, Estimates for twisted convolution operators with singularities of kernels on a sphere and at the origin, Differ. Uravn., 42, No 5 (2006), 674-683.
  46. [46] A.N. Karapetyants, V.A. Nogin, Inversion of potential-type operators with kernels possessing singularities on the sphere, Journal of Contemporary Math. Anal., 34, No 1 (1999), 55-69. 500 E.L. Shishkina
  47. [47] S.N. Askhabov, Nonlinear integral equations with kernels of potential type on a segment, In: Proc. of the Seventh International Conference on Differential and Functional-Differential Equations (Moscow, August 2229, 2014). Part 3, CMFD, 60, PFUR, M. (2016), 5-22.
  48. [48] I.A. Kipriyanov and M.I. Klyuchantsev, On singular integrals generated by generalised translation, II. Sibirskij Matematiceskij Zurnal, 11, No 5 (1970), 1060-1083.
  49. [49] L.N. Lyakhov, On one class of hypersingular integrals, Dokl. Acad. Nauk USSR, 315, No 2 (1990), 291296.
  50. [50] L.N. Lyakhov, Description of Riesz B-potential spaces Uα γ (Lγp ) using B-derevative of order 2[α/2], Dokl. Acad. Nauk USSR, 341, No 2 (1995), 161-165.