There are uniform constants $\bar{v}_{0}>0$ and $\underline{v}_{0}>0$ such that for each $\bm{x}\in M_{T}$, the $C^{k,\alpha}$-regularity scale $r_{k,\alpha}(\bm{x})$ at $\bm{x}$ has an explicit bound

(4.268)

$$\underline{v}_{0}\leq\frac{r_{k,\alpha}(\bm{x})}{\mathfrak{s}(\bm{x})}\leq\bar{v}_{0}.$$

The scale function $\mathfrak{s}(\bm{x})$ is expressed as follows,

(4.269)

$\displaystyle\mathfrak{s}(\bm{x})=(\frac{L_{T}(\bm{x})}{T})^{\frac{1}{2}}\cdot\mathfrak{r}(\bm{x})\cdot T^{\frac{1}{n}},\quad\bm{x}\in\mathcal{M}_{T},$

where $L_{T}(\bm{x})$ is defined in (4.12). Moreover, $k=2$ in Region $\bf{I}_{1}$. In all other cases, $k$ is any positive integer.