Holomorphic Sobolev spaces and the generalized Segal-Bargmann transform

摘要

We consider the generalized Segal-Bargmann transform C, 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 K-C to be in the image under C-t of C-infinity (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. (C) 2004 Elsevier Inc. All rights reserved.