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We show that the quotient of two caloric functions which vanish on a portionof the lateral boundary of a $H^{k+ \alpha}$ domain is $H^{k+ \alpha}$ up tothe boundary for $k \geq 2$. In the case $k=1$, we show that the quotient is in$H^{1+\alpha}$ if the domain is assumed to be space-time $C^{1, \alpha}$regular. This can be thought of as a parabolic analogue of a recent importantresult in [DS1], and we closely follow the ideas in that paper. We also givecounterexamples to the fact that analogous results are not true at points onthe parabolic boundary which are not on the lateral boundary, i.e., pointswhich are at the corner and base of the parabolic boundary.

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