Harnack inequality for singular or degenerate parabolic equations in non-divergence form

This paper studies a class of linear parabolic equations in non-divergence form in which the leading coefficients are measurable and they can be singular or degenerate as a weight belonging to the $A_{1+\frac{1}{n}}$ class of Muckenhoupt weights. Krylov-Safonov Harnack inequality for solutions is proved under some smallness assumption on a weighted mean oscillation of the weight. The novelty of the paper is an introduction of a class of generic weighted parabolic cylinders in which several growth lemmas are proved. A perturbation method is used and the Aleksandrov-Bakelman-Pucci type maximum principle for parabolic operators is crucially applied to suitable barrier functions to prove the growth lemmas. As corollaries, H\"{o}lder regularity estimates of solutions with respect to a quasi-distance, and a Liouville type theorem are obtained in the paper.