  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.
