The weighted Hölder space is defined by,

(8.2)

$$\|\omega\|_{C_{\delta,\nu,\mu}^{k,\alpha}(\mathcal{M})}\equiv\sum\limits_{j=0}^{k}\Big\|\rho_{\delta,\nu,\mu}^{(j)}\cdot\nabla^{j}\omega\Big\|_{C^{0}(\mathcal{M})}+[\omega]_{C_{\delta,\nu,\mu}^{k,\alpha}(\mathcal{M})},$$

where

(8.3)

$$[\omega]_{C_{\delta,\nu,\mu}^{k,\alpha}(\mathcal{M})}\equiv\sup_{\begin{subarray}{c}d_{g}(x,y)\leq r_{0}\\ x,y\in\mathcal{M}\end{subarray}}\Big\{\min\{\rho_{\delta,\nu,\mu}^{(k+\alpha)}(x),\rho_{\delta,\nu,\mu}^{(k+\alpha)}(y)\}\cdot\frac{|\nabla^{k}\omega(x)-\nabla^{k}\omega(y)|}{(d_{g}(x,y))^{\alpha}}\Big\},$$

$\omega\in T^{r,s}(\mathcal{M})$ is a tensor field of type $(r,s)$ and $r_{0}\equiv\frac{1}{2}\InjRad_{g}(\mathcal{M})$. In the above definition, the difference of the two covariant derivatives is defined in terms of the parallel translation.