Hardy spaces of heat functions

Abstract:

We consider spaces of solutions of the one-dimensional heat equation on appropriate bounded domains in the $(x, t)$-plane. The domains we consider have the property that they are parabolically star-shaped at some point; i.e., each downward half-parabola from some center point intersects the boundary exactly once. We introduce parabolic coordinates $(r, \theta )$ in such a way that the curves $\theta =\text {constant}$ are the half-parabolas, and dilation by multiplying by $r$ preserves heat functions. An integral kernel is introduced by specializing to this situation the very general kernel developed by Gleason and the author for abstract harmonic functions. The combination of parabolic coordinates and kernel function provides a close analogy with the Poisson kernel and polar coordinates for harmonic functions on the disc, and many of the Hardy space theorems for harmonic functions generalize to this setting. Moreover, because of the generality of the Bear-Gleason kernel, much of this theory extends nearly verbatim to other situations where there are polar-type coordinates (such that the given space of functions is preserved by the “radial” expansion) and the maximum principle holds. For example, most of these theorems hold unchanged for harmonic functions on a radial star in ${\mathbf {R}^n}$. As ancillary results we give a simple condition that a boundary point of a plane domain be regular, and give a new local Phragmén-Lindelöf theorem for heat functions.