Revista de la
Unión Matemática Argentina
Segal–Bargmann transforms and holomorphic Sobolev spaces of fractional order
Sundaram Thangavelu
Volume 66, no. 1 (2023), pp. 297–310    

https://doi.org/10.33044/revuma.4363

Download PDF

Abstract

In this note we investigate the image of Sobolev spaces of fractional order on a compact Lie group $K$ under the Segal–Bargmann transform. We show that the image can be characterised in terms of certain weighted Bergman spaces of holomorphic functions on the complexification extending a theorem of Hall and Lewkeeratiyutkul. We also treat the heat kernel transform associated to the Hermite operator.

References

  1. D.-W. Byun, Inversions of Hermite semigroup, Proc. Amer. Math. Soc. 118 no. 2 (1993), 437–445.  DOI  MR  Zbl
  2. L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 no. 7-9 (2007), 1245–1260.  DOI  MR  Zbl
  3. N. Garofalo, Fractional thoughts, in New Developments in the Analysis of Nonlocal Operators, Contemp. Math. 723, Amer. Math. Soc., Providence, RI, 2019, pp. 1–135.  DOI  MR  Zbl
  4. B. C. Hall, The Segal-Bargmann “coherent state” transform for compact Lie groups, J. Funct. Anal. 122 no. 1 (1994), 103–151.  DOI  MR  Zbl
  5. B. C. Hall and W. Lewkeeratiyutkul, Holomorphic Sobolev spaces and the generalized Segal-Bargmann transform, J. Funct. Anal. 217 no. 1 (2004), 192–220.  DOI  MR  Zbl
  6. B. Krötz, G. Ólafsson, and R. J. Stanton, The image of the heat kernel transform on Riemannian symmetric spaces of the noncompact type, Int. Math. Res. Not. no. 22 (2005), 1307–1329.  DOI  MR  Zbl
  7. B. Krötz, S. Thangavelu, and Y. Xu, The heat kernel transform for the Heisenberg group, J. Funct. Anal. 225 no. 2 (2005), 301–336.  DOI  MR  Zbl
  8. R. Radha and S. Thangavelu, Holomorphic Sobolev spaces, Hermite and special Hermite semigroups and a Paley-Wiener theorem for the windowed Fourier transform, J. Math. Anal. Appl. 354 no. 2 (2009), 564–574.  DOI  MR  Zbl
  9. M. B. Stenzel, The Segal-Bargmann transform on a symmetric space of compact type, J. Funct. Anal. 165 no. 1 (1999), 44–58.  DOI  MR  Zbl
  10. P. R. Stinga and J. L. Torrea, Extension problem and Harnack's inequality for some fractional operators, Comm. Partial Differential Equations 35 no. 11 (2010), 2092–2122.  DOI  MR  Zbl
  11. S. Thangavelu, Holomorphic Sobolev spaces associated to compact symmetric spaces, J. Funct. Anal. 251 no. 2 (2007), 438–462.  DOI  MR  Zbl
  12. S. Thangavelu, An analogue of Gutzmer's formula for Hermite expansions, Studia Math. 185 no. 3 (2008), 279–290.  DOI  MR  Zbl
  13. S. Thangavelu, Gutzmer's formula and Poisson integrals on the Heisenberg group, Pacific J. Math. 231 no. 1 (2007), 217–237.  DOI  MR  Zbl