Holomorphic Sobolev spaces and the generalized Segal–Bargmann transform

Abstract

We consider the generalized Segal-Bargmann transform C t for a compact group K; introduced in Hall (J. Funct. Anal. 122 (1994) 103). 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 in the image under C t of C N ðKÞ: 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.

Holomorphic Sobolev spaces associated to compact symmetric spaces

“…By using good estimates on the heat kernel on noncompact Riemannian symmetric spaces, recently proved by Anker and Ostellari [3], we obtain necessary and sufficient conditions on a holomorphic function to be in the image of C ∞ (X). This extends the result of Hall and Lewkeeratiyutkul [12] to all compact symmetric spaces. We also characterise the image of distributions under the heat kernel transform, settling a conjecture stated in [12].…”

“…This extends the result of Hall and Lewkeeratiyutkul [12] to all compact symmetric spaces. We also characterise the image of distributions under the heat kernel transform, settling a conjecture stated in [12]. The results in Section 4 depend on the characterisation of holomorphic Sobolev spaces in terms of the holomorphic Fourier coefficients of a function.…”

“…This happens precisely when we are dealing with compact Lie groups as symmetric spaces. In this case, we have explicit formulas even for derivatives of the heat kernel and this has been made use of by Hall and Lewkeeratiyutkul [12] in their study of holomorphic Sobolev spaces associated to compact Lie groups. In 2003, Anker and Ostellari [3] has sketched a proof for the following estimate for the heat kernel γ 1 t .…”

“…Recently, some surprising results came out on Heisenberg groups (see Krötz, Thangavelu, Xu [14]) and Riemannian symmetric spaces of noncompact type (see Krötz,Ólafsson,Stanton [15]). In 2004, Hall and Lewkeeratiyutkul [12] considered the Segal-Bargmann transform on Sobolev spaces H 2m (G) on compact Lie groups. They have shown that the image can be characterised as certain holomorphic Sobolev spaces.…”

Toeplitz Operators with Special Symbols on Segal–Bargmann Spaces

“…Finally, in the last section we consider Toeplitz operators on Segal-Bargmann spaces associated to compact Lie groups and symmetric spaces. For results closely related to the theme of this paper we refer to [2,[10][11][12].…”