For each $\delta\in\mathbb{R}$, $k\in\mathbb{Z}_{\geq 0}$ and $\alpha\in(0,1)$ define the weighted Hölder norm $C^{k,\alpha}_{\delta}$ by

$$\|a\|_{C^{k,\alpha}_{\delta}}=\sum_{j=1}^{k}{\|\rho_{\epsilon}^{-\delta+j}\nabla^{j}a\|_{C^{0}}}+\text{sup}_{d(x,y)<\text{inj}\,g_{\epsilon}}{\min{\left\{\rho_{\epsilon}(x)^{-\delta+k+\alpha},\rho_{\epsilon}(y)^{-\delta+k+\alpha}\right\}}\frac{|\nabla^{k}a(x)-\nabla^{k}a(y)|}{|x-y|^{\alpha}}}.$$

Here all norms and covariant derivatives are computed with respect to the metric $g_{\epsilon}$ and $\nabla^{k}a(x)$ and $\nabla^{k}a(y)$ are compared using parallel transport along the unique geodesic connecting $x$ and $y$. Similarly set $\|a\|_{C^{0}_{\delta}}=\|\rho^{-\delta}a\|_{C^{0}}$.