Segal–Bargmann transforms and holomorphic Sobolev spaces of fractional order
DOI:
https://doi.org/10.33044/revuma.4363Abstract
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.
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D.-W. Byun, Inversions of Hermite semigroup, Proc. Amer. Math. Soc. 118 no. 2 (1993), 437–445. DOI MR Zbl
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
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
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
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
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
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
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
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
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
S. Thangavelu, Holomorphic Sobolev spaces associated to compact symmetric spaces, J. Funct. Anal. 251 no. 2 (2007), 438–462. DOI MR Zbl
S. Thangavelu, An analogue of Gutzmer's formula for Hermite expansions, Studia Math. 185 no. 3 (2008), 279–290. DOI MR Zbl
S. Thangavelu, Gutzmer's formula and Poisson integrals on the Heisenberg group, Pacific J. Math. 231 no. 1 (2007), 217–237. DOI MR Zbl
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