We just need to establish the following:

1. (1)

   (General derivatives estimate) For each $k\in\mathbb{N}$ and $\epsilon>0$, it holds that

   (3.211)

   $$|\nabla^{k+2}T|=O(r^{-(k+\epsilon)}).$$

2. (2)

   (Mixed derivatives estimate) For each $k\in\mathbb{N}$, $\ell\in\mathbb{N}$ and $\epsilon>0$, it holds that

   (3.212)

   $$|(\nabla^{t})^{k}\nabla^{\ell+2}T|=O(r^{-(\ell+\epsilon)}),$$

   where $\nabla^{t}$ denotes the tangential derivative.

The above estimates will be proved by induction.

First, we will prove the following order estimate for $\nabla^{2}T$ in a smaller neighborhood $\mathcal{U}^{\prime}\subset\subset\mathcal{U}$

(3.213)

$$|\nabla^{2}T|=O(r^{-\epsilon}).$$

This can be viewed as the base step for carrying out the inductive argument.

To begin with, by definition, for any $p>0$, we have $|v|_{L^{p}(\mathcal{U})}\leq C_{p}$. Applying the standard elliptic $W^{2,p}$-estimate, for each $p>0$, there is some constant $C_{p}>0$ such that in a smaller neighborhood $\mathcal{U}_{1}\subset\subset\mathcal{U}$ such that

(3.214)

$$\|\nabla^{2}T\|_{L^{p}(\mathcal{U}_{1})}\leq C_{p}.$$

Then Sobolev embedding theorem tells us that

(3.215)

$$T\in W^{2,p}(\mathcal{U}_{1})\cap C^{1,\alpha}(\mathcal{U}_{1})$$

for any $p>1$ and $0<\alpha<1$.

To prove (3.213), we need to differentiate the equation, which schematically yields that

(3.216)

$$\Delta\nabla^{2}T+Q_{1}*\nabla^{3}T+Q_{2}*\nabla^{2}T+Q_{3}*\nabla T=w,$$

where $|w|=O^{\prime}(r^{-2})$ and $Q_{i}$’s are smooth terms arising from differentiating the coefficients of $\Delta$. The above equation can be viewed as an elliptic system in terms of the Hessian of $T$. Let $\Phi\equiv\nabla^{2}T$, noticing $\nabla T\in C^{\alpha}(\mathcal{U}_{1})$, so the terms involving $\nabla T$ can be absorbed to the right hand side of the equation. Then $\Phi$ can be treated as vector valued functions, once we fix a local frame. So it follows that

(3.217)

$$\Delta\Phi+Q_{1}*\nabla\Phi+Q_{2}*\Phi=\eta,$$

where $|\eta|=O^{\prime}(r^{-2})$. Since $\eta$ has unbounded $L^{p}$-norm for large $p$, the standard $W^{2,p}$-estimate for $\Phi$ does not directly apply.

For improving the regularity of $\nabla^{2}T$, we will rescale the metric $g$. For each $x$ in an even smaller neighborhood $\mathcal{U}_{2}$ with $r(x)=r_{x}$, we rescale the metric $g$ in $B_{r_{x}}(x)$ by letting

(3.218)

$$\tilde{g}=(r_{x})^{-2}g,$$

then the following equation holds in the rescaled geodesic ball $B_{1}^{\tilde{g}}(x)$,

(3.219)

$$\widetilde{\Delta}\widetilde{\Phi}+r_{x}\widetilde{Q}_{1}*\widetilde{\nabla}\widetilde{\Phi}+(r_{x})^{2}\widetilde{Q}_{2}*\widetilde{\Phi}=\tilde{\eta},\quad|\tilde{\eta}|=O^{\prime}(1),$$

where $\widetilde{\Phi}(y)=\Phi(r_{x}\cdot y)$ for each $y\in B_{1}^{\tilde{g}}(x)$. In the above equation, all the coefficients are uniformly bounded independent of $x$. Since we have shown $|\Phi|_{L^{p}}\leq C_{p}$ in (3.214), so simple rescaling gives rise to the following estimate for any $p>0$,

(3.220)

$$|\widetilde{\Phi}|_{L^{p}(B_{1}^{\tilde{g}}(x))}\leq C_{p}\cdot(r_{x})^{-\frac{n}{p}}.$$

Now applying the $W^{2,p}$-estimate for $\widetilde{\Phi}$, then for each $p>0$

(3.221)

$$|\widetilde{\Phi}|_{W^{2,p}(B_{1/2}^{\tilde{g}}(x))}\leq C_{p}\cdot(r_{x})^{-\frac{n}{p}}$$

with $C_{p}>0$ independent of $x$. By the Sobolev embedding

(3.222)

$$|\widetilde{\Phi}|_{C^{1,\alpha}(B_{1/4}^{\tilde{g}}(x))}\leq C_{\alpha,p}\cdot(r_{x})^{-\frac{n}{p}}$$

with $C_{\alpha,p}>0$ independent of $x$. Scale back to the original metric, for any $p>0$, there is some $C_{\alpha,p}>0$ independent of the base point $x\in\mathcal{U}_{2}$ such that

(3.223)

$$|\Phi|_{L^{\infty}(B_{r_{x}/4}(x))}+|r_{x}\nabla\Phi|_{L^{\infty}(B_{r_{x}/4}(x))}\leq C_{\alpha,p}\cdot(r_{x})^{-\frac{n}{p}}.$$

This completes the proof of (3.213).

Now we will finish the proof of Item (1) by using the induction. Based on (3.223), the key induction step is to prove the following: Given any $\ell\in\mathbb{Z}_{+}$, if for each $\epsilon>0$ and $0\leq k\leq\ell-1$,

(3.224)

$$|\nabla^{k}\Phi|=O(r^{-(k+\epsilon)}),\quad|\widetilde{\nabla}^{k}\widetilde{\Phi}|_{W^{2,p}(B_{1/2}^{\tilde{g}}(x))}\leq C_{k,p,\epsilon}\cdot(r_{x})^{-\epsilon},$$

then for each $\epsilon>0$, we have

(3.225)

$$|\nabla^{\ell}\Phi|=O(r^{-(\ell+\epsilon)}),\quad|\widetilde{\nabla}^{\ell}\widetilde{\Phi}|_{W^{2,p}(B_{1/4}^{\tilde{g}}(x))}\leq C_{\ell,p,\epsilon}\cdot(r_{x})^{-\epsilon}.$$

Indeed, then differentiating (3.216) by $\nabla^{k}$,

(3.226)

$$\Delta(\nabla^{\ell}\Phi)+\sum\limits_{j=1}^{\ell+1}Q_{j}*\nabla^{j}\Phi=w_{\ell},$$

where $|w_{\ell}|=O^{\prime}(r^{-(\ell+2)})$. As before, we rescale the metric $g$ by taking $\tilde{g}=(r_{x})^{-2}g$, then

(3.227)

$$\widetilde{\Delta}(\widetilde{\nabla}^{\ell}\widetilde{\Phi})+\sum\limits_{j=1}^{\ell+1}(r_{x})^{\ell-j+2}\cdot Q_{j}*\nabla^{j}\widetilde{\Phi}=\tilde{w}_{\ell},$$

where $|\tilde{w}_{\ell}|=O^{\prime}(1)$. Let $k=\ell-1$, applying the induction hypothesis (3.224) and Sobolev embedding, we have

(3.228)

$$|\widetilde{\nabla}^{\ell}\widetilde{\Phi}|_{L^{\infty}(B_{1/2}^{\tilde{g}}(x))}=O((r_{x})^{-\epsilon}).$$

The above enables us to apply the $W^{2,p}$-elliptic estimate, so we obtain the following estimate for each $\epsilon>0$,

(3.229)

$$|\widetilde{\nabla}^{\ell}\widetilde{\Phi}|_{W^{2,p}(B_{1/4}^{\tilde{g}}(x))}\leq C_{\ell,p,\epsilon}\cdot(r_{x})^{-\epsilon},$$

where $C_{\ell,p,\epsilon}>0$ is independent of the base point $x$. Applying the Sobolev embedding $W^{2,p}\subset C^{1,\alpha}$ and scaling back to the original metric $g$,

(3.230)

$$|\nabla^{\ell}\Phi|_{L^{\infty}(B_{r_{x}/8}(x))}\leq C_{\ell,\epsilon}\cdot(r_{x})^{-(\ell+\epsilon)}$$

for each $\epsilon>0$. So we complete the proof of Item (1).

Now we are ready to finish the proof of Item (2). We only focus on the case $\ell=0$ and the case for $\ell>0$ can be directly achieved by applying the above rescaling arguments. To this end, we need the following claim for the tangential derivatives estimate.

Claim. Let $|\widetilde{\Phi}|_{W^{2,p}(B_{1/2}^{\tilde{g}}(x))}\leq C_{p,\epsilon}\cdot(r_{x})^{-\epsilon}$ for any $\epsilon>0$ and for any $x\in\mathcal{U}$. Assume that $\widetilde{\Phi}$ solves the elliptic equation

(3.231)

$$\Delta\widetilde{\Phi}+Q_{1}*\nabla\widetilde{\Phi}+Q_{2}*\widetilde{\Phi}=\zeta,$$

where $Q_{i}$’s are smooth coefficients, $|\zeta|\in O^{\prime}(1)$. Then for any $x\in\mathcal{U}$, the estimate

(3.232)

$$|(\nabla^{t})^{k}\widetilde{\Phi}|_{W^{2,p}(B_{1/8}^{\tilde{g}}(x))}\leq C_{k,p,\epsilon}\cdot(r_{x})^{-\epsilon},$$

holds for all $k\geq 1$, $p>1$ and $\epsilon>0$.

Taking the first tangential derivative $\nabla^{t}$ for $\Delta\widetilde{\Phi}$,

(3.233)

$$(\nabla^{t})(\Delta\widetilde{\Phi})=\Delta\nabla^{t}\widetilde{\Phi}+Q_{1}*\nabla^{2}\widetilde{\Phi}+Q_{2}*\nabla\widetilde{\Phi},$$

where $Q_{i}$’s are smooth functions. Hence differentiating (3.231) once by the tangential derivative $\nabla^{t}$, we have

(3.234)

$$\Delta(\nabla^{t}\widetilde{\Phi})+Q_{1}*\nabla^{2}\widetilde{\Phi}+Q_{2}*\nabla\widetilde{\Phi}=w_{1},$$

where $Q_{i}$’s are smooth functions, $w_{1}\equiv\nabla^{t}\zeta$ and $|w_{1}|\in O^{\prime}(1)$. Since we have already assumed $|\nabla^{2}\widetilde{\Phi}|_{L^{p}(B_{1/2}^{\tilde{g}}(x))}\leq C_{p,\epsilon}\cdot(r_{x})^{-\epsilon}$ for all $\epsilon>0$, applying the standard $W^{2,p}$-estimate, then for any $p>1$ and $\epsilon>0$,

(3.235)

$$|\nabla^{t}\widetilde{\Phi}|_{W^{2,p}(B_{1/3}^{\tilde{g}}(x))}\leq C_{p,\epsilon}\cdot(r_{x})^{-\epsilon}.$$

Now we prove the higher order mixed derivatives estimate by induction. Repeat taking the tangential derivatives and let $\Psi^{(k)}\equiv(\nabla^{t})^{k}\widetilde{\Phi}$ for all $k>1$. Assume that $|\Psi^{(j)}|_{W^{2,p}(B_{1/4}^{\tilde{g}}(x))}\leq C_{k,p,\epsilon}(r_{x})^{-\epsilon}$ holds for all $j\leq k-1$ and $p>1$, then

(3.236)

$$\Delta(\Psi^{(k)})+\sum_{\begin{subarray}{c}\mu=1,2\\ 1\leq\mu+\nu\leq k\end{subarray}}Q_{\mu\nu}*\nabla^{\mu}(\Psi^{(\nu)})=w_{k},$$

where $w_{k}\equiv(\nabla^{t})^{k}\zeta$ and $|w_{k}|=O^{\prime}(1)$. Applying the induction hypothesis, it follows that for any $k\geq 1$,

(3.237)

$$|\Delta(\Psi^{(k)})|_{L^{p}(B_{1/4}^{\tilde{g}}(x))}\leq C_{k,p,\epsilon}\cdot(r_{x})^{-\epsilon}$$

Therefore, for any $k\geq 1$, $p>1$ and $\epsilon>0$, there is some constant $C_{k,p,\epsilon}>0$ such that

(3.238)

$$|(\nabla^{t})^{k}\widetilde{\Phi}|_{W^{2,p}(B_{1/8}^{\tilde{g}}(x))}=|\Psi^{(k)}|_{W^{2,p}(B_{1/8}^{\tilde{g}}(x))}\leq C_{k,p,\epsilon}\cdot(r_{x})^{-\epsilon}.$$

This completes the proof of the claim.

Now we are in a position to finish the proof of the lemma by completing the induction arguments for Item (2). As before, for any $x$, under the rescaled metric $\tilde{g}=(r_{x})^{-2}g$, we start with the equation for $\widetilde{\Phi}$ in the rescaled geodesic ball $B_{1}^{\tilde{g}}(x)$,

(3.239)

$$\widetilde{\Delta}\widetilde{\Phi}+r_{x}\widetilde{Q}_{1}*\widetilde{\nabla}\widetilde{\Phi}+(r_{x})^{2}\widetilde{Q}_{2}*\widetilde{\Phi}=\tilde{\eta},\ 0<r_{x}<1,$$

where $\widetilde{Q}_{i}$’s are smooth functions and $|\tilde{\eta}|=O^{\prime}(1)$. The above claim tells us that for any $k\geq 1$, $p>1$ and $\epsilon>0$,

(3.240)

$$|(\nabla^{t})^{k}\widetilde{\Phi}|_{W^{2,p}(B_{1/2}^{\tilde{g}}(x))}\leq C_{k,p,\epsilon}\cdot(r_{x})^{-\epsilon},$$

where $C_{k,p,\epsilon}>0$ is independent of $x$. Applying the Sobolev embedding, then for any $0<\alpha<1$,

(3.241)

$$|(\nabla^{t})^{k}\widetilde{\Phi}|_{C^{1,\alpha}(B_{1/4}^{\tilde{g}}(x))}\leq C_{k,\alpha,\epsilon}\cdot(r_{x})^{-\epsilon}.$$

In particular, $|(\nabla^{t})^{k}\widetilde{\Phi}|=O(r^{-\epsilon})$ for all $\epsilon>0$. Notice that the above estimate is independent of the choice of $x$. Rescaling back to the original metric, then for each $\epsilon>0$,

(3.242)

$$|(\nabla^{t})^{k}\nabla^{2}T|=O(r^{-\epsilon}).$$

The proof of the lemma is done.

∎