Minimal thickness and uniqueness of kernel functions for the heat equation in several variables (Q1343886)

A kernel function for the heat equation on a domain \(\Omega\) in \(\mathbb{R}^{n + 1}\) is a nonnegative temperature which vanishes continuously on the parabolic boundary of \(\Omega\), except at a given point which may be at infinity, but does not vanish identically. Uniqueness of such functions (up to a multiplicative constant) is established for certain domains of the form \(\Omega = \{(x,t) : T > t > \varphi (x)\}\), and their images under Appell transformation, where \(T\) is a positive constant and \(\varphi\) a nonnegative, upper semicontinuous function.