Segal-Bargmann transform and holomorphic Sobolev spaces of fractional order

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.