THE HEAT TRANSFORM ON THE COMPLEX PLANE

Abstract

The heat transform $H_t$, for positive time , is the convolution on the complex plane with the heat kernel. In the field of analytic function spaces and related operator theory, $H_t$ coincides with the Berezin transform for the Fock space $F_t^2$ induced by the Gaussian measure $e^{-|z{|}^2∕t}dA(z)$. We study fixed-points of $H_t$ and the limit behavior of $H_{t}f$ as $t\to 0^{+}$. Fixed-points of $H_t$ are shown to be closely related to eigenfunctions of the Laplacian corresponding to certain special eigenvalues, while the limit behavior of $H_{t}f$ as $t\to 0^{+}$ 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 $H_t$ are known, but we present a completely new proof here.