The image of the heat kernel transform on Riemannian symmetric spaces of   the noncompact type
  The heat kernel or Bargmann-Segal transform on a noncompact Riemannian symmetric space X=G/K maps a square integrable function on X to a holomorphic function on the complex crown. In this article we determine the range of this transform. 
Bernhard Kroetz, Gestur Olafsson, Robert Stanton