The range of the heat operator
  This paper describes results characterizing the range of the time-t heat operator on various manifolds, including Euclidean spaces, spheres, and hyperbolic spaces. The guiding principle behind these results is this: The functions in the range of the heat operator should be, roughly, those functions having an analytic continuation to the appropriate complexified manifold with growth in the imaginary directions at most like that of the time-t heat heat kernel. 
Brian C. Hall