Holomorphic Sobolev spaces and the generalized Segal-Bargmann transform

TL;DR

This paper characterizes the image of the generalized Segal-Bargmann transform on compact groups, providing growth conditions for holomorphic functions to correspond to smooth or Sobolev functions on the group.

Contribution

It offers necessary and sufficient growth conditions for holomorphic functions to be images of smooth and Sobolev functions under the transform, using a holomorphic Sobolev embedding theorem.

Findings

Characterization of the image of C_t for smooth functions on K

Growth conditions for holomorphic functions to be in the transform's image

Holomorphic Sobolev embedding theorem applied to the transform

Abstract

We consider the generalized Segal-Bargmann transform C_t for a compact group K, introduced in B. C. Hall, J. Funct. Anal. 122 (1994), 103-151. 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 the image under C_t of a C-infinity function on 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.