Holomorphic Sobolev spaces and the generalized Segal–Bargmann transform

2004/06/16 by Brian C. Hall, Wicharn Lewkeeratiyutkul · 1 citation
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Holomorphic and Operator Theory #Mathematical Analysis and Transform Methods #math-ph #math.MP #msc:22E30 #msc:81S30

paper · pdf · doi:10.1016/j.jfa.2004.03.018

published as J. Functional Analysis 217 (2004) 192--220 · To appear in J. Functional Analysis

arxiv created 2004/06/16 · openalex publication_date 2004/07/06 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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• Holomorphic Sobolev spaces, Hermite ans special Hermite semigroups and a Paley-Wiener theorem for the windowed Fourier transform

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