We establish the Alexandroff-Bakelman-Pucci estimate, the Harnack inequality, the H\"older regularity and the Schauder estimates to a class of degenerate parabolic equations of non-divergence form in all dimensions \begin{ equation} \mathcal{ L}u:= u_t -Lu= u_t -(x a_{ 11} u_{ xx} +2\sqrt{ x} \sum_{ j=2}^n a_{ 1j} u_{ x y_j} + \sum_{ i,j=2}^n a_{ ij} u_{ y_i y_j} + b_1 u_x +\sum_{ j=2}^n b_j u_{ y_j} ) =g\ \end{ equation} on \(x \geq 0, y=(y_2,\ldots, y_n) \in \mathbb{ R}^{ n-1}\), with bounded measurable coefficients.
Categories: math.AP math.DG