Hölder regularity of solutions of degenerate parabolic equations of general dimension

We establish the Alexandroff-Bakelman-Pucci estimate, the Harnack inequality, the Hölder 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 $x\ge 0,y=(y_2,\dots ,y_n)\in {\mathbb{R}}^{n-1}$, with bounded measurable coefficients.