Segal-Bargmann transform and holomorphic Sobolev spaces of fractional order

TL;DR

This paper characterizes the image of fractional Sobolev spaces on compact Lie groups under the Segal-Bargmann transform, linking it to weighted Bergman spaces and extending previous theorems.

Contribution

It extends the understanding of the Segal-Bargmann transform to fractional Sobolev spaces on compact Lie groups and relates it to holomorphic function spaces.

Findings

Characterization of the image of fractional Sobolev spaces under the transform

Extension of Hall and Lewkeeratiyutkul's theorem to fractional orders

Analysis of the heat kernel transform for the Hermite operator

Abstract

In this note we investigate the image of Sobolev spaces of fractional order on a compact Lie group $ K $ under the Segal-Bargmann transform. We show that the image can be characterised in terms of certain weighted Bergman spaces of holomorphic functions on the complexification extending a theorem of Hall and Lewkeeratiyutkul. We also treat the heat kernel transform associated to the Hermite operator.