Holomorphic Sobolev spaces and the generalized Segal-Bargmann transform
Abstract
We consider the generalized Segal-Bargmann transform Ct for a compact group K, introduced in B. C. Hall, J. Funct. Anal. 122 (1994), 103-151. Let KC denote the complexification of K. We give a necessary-and-sufficient pointwise growth condition for a holomorphic function on KC to be the image under Ct of a C-infinity function on 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.
0
Turn this paper into a full lesson
ArcXiv compiles a staged curriculum from this paper: 8-12 lessons across beginner → advanced, synthesised section guides, visuals, flashcards, a quiz, exercises, and on-demand deep dives per section. Grounded in the abstract, never invented.