Complex Analysis and Operator Theory

, Volume 12, Issue 3, pp 669–681 | Cite as

The Kontorovich–Lebedev Transform and Sobolev Type Space



In this paper boundedness of convolution operator in \(L^p(\mathbb {R}_+; ~xdx)\) is given and continuity of pseudo-differential operator \(\mathcal {P}_a\) associated with the Kontorovich–Lebedev transform (KL-transform) from \(H(\mathbb {R}_+)\) into itself is discussed. The Kontorovich–Lebedev potential (KL-potential) \(\mathcal {P}_a^s\) is defined on \(H(\mathbb {R}_+)\) and then it is extended to the space of distributions also some of its properties are studied. A Sobolev type space \(W^{s,p}(\mathbb {R}_+)\) associated with the KL-transform is discussed and proved as a Banach space. Moreover, it is shown that the KL-potential is an isometry of \(W^{s,p}\). An \(L^p\)-boundedness for the KL-potential is obtained and at the end applications of the KL-potential in pseudo-differential equation are discussed.


Kontorovich–Lebedev transform Convolution Pseudo-differential operator Banach space 

Mathematics Subject Classification

44A20 35S05 46F12 46E35 


  1. 1.
    Sneddon, I.N.: The Use of Integral Transforms. McGraw-Hill Book Co., New York (1972)MATHGoogle Scholar
  2. 2.
    Banerji, P.K., Loonker, D., Kalla, S.L.: Kontorovich–Lebedev transform for Boehmians. Integral Transforms Spec. Funct. 20(12), 905–913 (2009)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Prasad, A., Mandal, U.K.: Two versions of pseudo-differential operators involving the Kontorovich–Lebedev transform in \(L^2(\mathbb{R}_+; x^{-1}dx)\). Forum Math. doi: 10.1515/forum-2016-0254
  4. 4.
    Prasad, A., Mandal, U.K.: Boundedness of pseudo-differential operators involving Kontorovich–Lebedev transform. Integral Transforms Spec. Funct. 28(4), 300–314 (2017)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Erde’lyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Tables of Integral Transforms, vol. 2. McGraw-Hill Book Co., New York (1953)Google Scholar
  6. 6.
    Zemanian, A.H.: The Kontorovich–Lebedev transformation on distributions of compact support and its inversion. Math. Proc. Camb. Phil. Soc. 77, 139–143 (1975)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Yakubovich, S.B.: Index Transforms. World Scientific, Singapore (1996)CrossRefMATHGoogle Scholar
  8. 8.
    Srivastava, H.M., González, B.J., Nergin, E.R.: New \(L^p\)-boundedness properties of the Kontorovich–Lebedev and Mehler–Fock transforms. Integral Transforms Spec. Funct. 27(10), 835–845 (2016)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    González, B.J., Negrín, E.R.: Operational calculi for Kontorovich–Lebedev and Mehler–Fock transforms on distributions with compact support. Rev. Colmbania Mat. 32(1), 81–92 (1998)MathSciNetMATHGoogle Scholar
  10. 10.
    Glaeske, H.J., Heß, A.: A convolution connected with the Kontorovich–Lebedev transform. Math. Z. 193(1), 67–78 (1986)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Pathak, R.S.: Pseudo-differential operator associated with the Kontorovich–Lebedev transform. Invest. Math. Sci. 5(1), 29–46 (2015)Google Scholar
  12. 12.
    Yakubovich, S.B.: The Kontorovich–Lebedev transformation on Sobolev type spaces. Sarajevo J. Math. 1(14), 211–234 (2005)MathSciNetMATHGoogle Scholar
  13. 13.
    Prudnikov, A.P., Brychkov, Y.A., Marichev, O.I.: Integrals and Series, Vol. I, Elementary Functions. Gordon and Breach Science Publisher, Amsterdam (1986)MATHGoogle Scholar
  14. 14.
    Yakubovich, S.B.: On the least values of \(L^p\)-norms for the Kontorovich–Lebedev transform and its convolution. J. Approx. Theory 131(2), 231–242 (2004)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Wong, M.W.: An Introduction to Pseudo-differential Operators, 3rd edn. World Scientific, Singapore (2014)CrossRefMATHGoogle Scholar
  16. 16.
    Pathak, R.S., Pandey, P.K.: Sobolev type spaces associated with Bessel operator. J. Math. Anal. Appl. 215(1), 95–111 (1997)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Pathak, R. S., Shrestha, K. K.: Generalized Sobolev type space. Proc. Nat. Acad. Sci. India, 73(A)(1), 75–92 (2003)Google Scholar
  18. 18.
    Salem, N.B., Dachraoui, A.: Sobolev type spaces associated with Jacobi differential operator. Integral Transforms Spec. Funct. 9(3), 163–184 (2000)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Pathak, R.S.: A Course in Distribution Theory and Application. Narosa Publishing House, New Delhi (2009)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Applied MathematicsIndian Institute of Technology (Indian School of Mines)DhanbadIndia

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