A parabolic analogue of the higher-order comparison theorem of De Silva and Savin

We show that the quotient of two caloric functions which vanish on a portion of the lateral boundary of a Hk+αHk+α domain is Hk+αHk+α up to the boundary for k≥2k≥2. In the case k=1k=1, we show that the quotient is in H1+αH1+α if the domain is assumed to be space-time C1,αC1,α regular. This can be thought of as a parabolic analogue of a recent important result in [8], and we closely follow the ideas in that paper. We also give counterexamples to the fact that analogous results are not true at points on the parabolic boundary which are not on the lateral boundary, i.e., points which are at the corner and base of the parabolic boundary.