On the images of Dunkl–Sobolev spaces under the Schrödinger semigroup associated to Dunkl operators

  • C. Sivaramakrishnan
  • D. Sukumar
  • D. Venku NaiduEmail author


In this article, we consider the Schrödinger semigroup related to the Dunkl–Laplacian \(\Delta _{\mu }\) (associated to finite reflection group G) on \(\mathbb {R}^n\). We characterize the image of \(L^2(\mathbb {R}^n, e^{u^2} h_{\mu }(u) du)\) under the Schrödinger semigroup as a reproducing kernel Hilbert space. We define Dunkl–Sobolev space in \(L^2(\mathbb {R}^n, e^{u^2} h_{\mu }(u) du)\) and characterize it’s image under the Schrödinger semigroup associated to \(G=\mathbb {Z}_2^n\) as a reproducing kernel Hilbert space up to equivalence of norms. Also we provide similar results for Schrödinger semigroup associated to Dunkl–Hermite operator.


Segal–Bargmann transform Schrödinger semigroup Weighted Bergman space Dunkl–Sobolev space 

Mathematics Subject Classification

Primary 46E35 Secondary 46F12 47D06 35B65 35J10 



The authors wish to thank G.B. Folland for giving clarification to their questions related to weighted Sobolev spaces. The first author thanks University Grant Commission, India for the financial support. We thank anonymous referee for thorough and careful review of our manuscript and helping us to improve the manuscript and to bring it to the present form.


  1. 1.
    Bargmann, V.: On a Hilbert space of analytic functions and an associated integral transform. Commun. Pure Appl. Math. 14, 187–214 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ben Said, S., Orsted, B.: Segal–Bargmann transforms associated with finite Coxeter groups. Math. Ann. 334(2), 281–323 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cholewinski, F.M.: Generalized Fock spaces and associated operators. SIAM J. Math. Anal. 15(1), 177–202 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Dai, F., Xu, Y.: Analysis on \(h\)-harmonics and Dunkl Transforms. Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser/Springer, Basel (2015)CrossRefzbMATHGoogle Scholar
  5. 5.
    Dunkl, C.F.: Differential-difference operators associated to reflection groups. Trans. Am. Math. Soc. 311(1), 167–183 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Hall, B.C.: The inverse Segal–Bargmann transform for compact Lie groups. J. Funct. Anal. 143(1), 98–116 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Hall, B.C., Lewkeeratiyutkul, W.: Holomorphic Sobolev spaces and the generalized Segal–Bargmann transform. J. Funct. Anal. 217(1), 192–220 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hayashi, N., Saitoh, S.: Analyticity and smoothing effect for the Schrödinger equation. Ann. Inst. H. Poincaré Phys. Théor. 52(2), 163–173 (1990)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Kawazoe, T., Mejjaoli, H.: Uncertainty principles for the Dunkl transform. Hiroshima Math. J. 40(2), 241–268 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Krötz, B., Thangavelu, S., Xu, Y.: The heat kernel transform for the Heisenberg group. J. Funct. Anal. 225(2), 301–336 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Krötz, B., Ólafsson, G., Stanton, R.J.: The image of the heat kernel transform on Riemannian symmetric spaces of the noncompact type. Int. Math. Res. Not. 2005(22), 1307–1329 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lebedev, N.N.: Special Functions and Their Applications, Revised Edition, Translated from the Russian and Edited by Richard A. Silverman. Dover Publications Inc, New York (1972)Google Scholar
  13. 13.
    Parui, S., Ratnakumar, P.K., Thangavelu, S.: Analyticity of the Schrödinger propagator on the Heisenberg group. Monatsh. Math. 168(2), 279–303 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Radha, R., Thangavelu, S.: Holomorphic Sobolev spaces, Hermite and special Hermite semigroups and a Paley–Wiener theorem for the windowed Fourier transform. J. Math. Anal. Appl. 354(2), 564–574 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Radha, R., Thangavelu, S., Naidu, D.V.: On the images of Sobolev spaces under the heat kernel transform on the Heisenberg group. Math. Nachr. 286(13), 1337–1352 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Rosenblum, M.: Generalized Hermite polynomials and the Bose-like oscillator calculus. In: Nonselfadjoint Operators and Related Topics (Beer Sheva, 1992). Operator Theory: Advances and Applications, vol. 73, pp. 369–396. Birkhäuser, BaselGoogle Scholar
  17. 17.
    Rosler, M.: Positivity of Dunkl’s intertwining operator. Duke Math. J. 98(3), 445–463 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Salem, N.B., Nefzi, W.: Inversion of the Dunkl–Hermite semigroup. Bull. Malays. Math. Sci. Soc. (2) 35(2), 287–301 (2012)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Salem, N.B., Nefzi, W.: Images of some functions and functional spaces under the Dunkl–Hermite semigroup. Comment. Math. Univ. Carolin. 54(3), 345–365 (2013)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Segal, I.E.: Mathematical problems of relativistic physics, With an appendix by George W. Mackey. Lectures in Applied Mathematics. In: Proceedings of the Summer Seminar, Boulder, Colorado, 1960, vol. II. American Mathematical Society, Providence (1963)Google Scholar
  21. 21.
    Soltani, F.: Generalized Fock spaces and Weyl commutation relations for the Dunkl kernel. Pac. J. Math. 214(2), 379–397 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Sontz, S.B.: The \(\mu \)-deformed Segal–Bargmann transform is a Hall type transform. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 12(2), 269–289 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Sontz, S.B.: On Segal–Bargmann analysis for finite Coxeter groups and its heat kernel. Math. Z. 269(1–2), 9–28 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Thangavelu, S., Xu, Y.: Convolution operator and maximal function for the Dunkl transform. J. Anal. Math. 97, 25–55 (2005)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology HyderabadHyderabadIndia

Personalised recommendations