Let $\mathcal{K}\subset\mathcal{M}_{T}$ be compact, then the weighted Hölder norm of a tensor field $\chi\in T^{r,s}(\mathcal{K})$ of type $(r,s)$ is defined by,

(4.285)		$\displaystyle\|\chi\|_{C_{\delta,\nu,\mu}^{k,\alpha}(\mathcal{K})}$	$\displaystyle\equiv\sum\limits_{m=0}^{k}\Big\|\rho_{\delta,\nu,\mu}^{(m)}\cdot\nabla^{m}\chi\Big\|_{C^{0}(\mathcal{K})}+[\chi]_{C_{\delta,\nu,\mu}^{k,\alpha}(\mathcal{K})},$
(4.286)		$\displaystyle[\chi]_{C_{\delta,\nu,\mu}^{k,\alpha}(\mathcal{K})}$	$\displaystyle\equiv\sup_{\begin{subarray}{c}d_{g}(x,y)\leq\iota_{0}\\ x,y\in\mathcal{K}\end{subarray}}\Big\{\min\{\rho_{\delta,\nu,\mu}^{(k+\alpha)}(x),\rho_{\delta,\nu,\mu}^{(k+\alpha)}(y)\}\cdot\frac{|\nabla^{k}\chi(x)-\nabla^{k}\chi(y)|}{(d_{g}(x,y))^{\alpha}}\Big\},$

where $\iota_{0}\equiv\frac{1}{4}\InjRad_{g}(\mathcal{M})$. In the above definition, the difference of the two covariant derivatives is defined in terms of the parallel translation along the minimal geodesic.