THE HEAT TRANSFORM ON THE COMPLEX PLANE

The heat transform Ht , for positive time t >0, is the convolution on the complex plane with the heat kernel. In the field of analytic function spaces and related operator theory, Ht coincides with the Berezin transform for the Fock space F2 t induced by the Gaussian measure e-|z|2/t d A(z). We study fixed-points of Ht and the limit behavior of Ht f as t →0+. Fixed-points of Ht are shown to be closely related to eigenfunctions of the Laplacian corresponding to certain special eigenvalues, while the limit behavior of Ht f as t →0+ depends on certain continuity and oscillation properties of f . The paper is expository, although it contains a few new results. In particular, the main results about fixed-points of Ht are known, but we present a completely new proof here.