Distributional image characterization for the generalized Segal–Bargmann transform

Let $K$ be the compact Lie group considered in the paper, let $K_{\mathbb C}$ be its complexification, let $t>0$, and let $C_t$ denote the generalized Segal–Bargmann transform from distributions on $K$ to holomorphic functions on $K_{\mathbb C}$. Write $x\in K$, $Y$ in the Lie algebra of $K$, and let $\Phi(Y)$ be the function appearing in the holomorphic growth estimates. Distributional image conjecture. A holomorphic function $F$ on $K_{\mathbb C}$ is of the form $F=C_tf$ for a distribution $f$ on $K$ if and only if there exist a positive integer $n$ and a constant $A>0$ such that

$$\left|F(xe^{iY})\right|^2\leq A\Phi(Y)e^{|Y|^2/t}(1+|Y|^2)^{2n}.$$

This conjecture seeks a precise growth characterization of the image of distributions under the generalized Segal–Bargmann transform, extending the corresponding results for $L^2$ functions and Sobolev spaces. The surrounding discussion states that the transform extends from $L^2(K)$ to distributions, but does not provide a resolution of this characterization.