Holomorphic Sobolev spaces and the generalized Segal–Bargmann transform

We consider the generalized Segal–Bargmann transform Ct for a compact group K, introduced in Hall (J. Funct. Anal. 122 (1994) 103). Let KC denote the complexification of K. We give a necessary-and-sufficient pointwise growth condition for a holomorphic function on KC to be in the image under Ct of C∞(K). We also characterize the image under Ct of Sobolev spaces on K. The proofs make use of a holomorphic version of the Sobolev embedding theorem