Holomorphic Sobolev spaces and the generalized Segal–Bargmann transform

Abstract

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