Since the metric $g$ is $C^{0}$ equivalent to the flat metric $g_{0}$, the Sobolev space $W^{1,p}$ is the same with respect to both metrics, and the Moser iteration works for the operator $\Delta=\Delta_{g}$. Here again we use the geometers’ convention for the sign. By Bochner formula there is a constant $C_{1}>0$ such that

$$\Delta|Rm|\leq C_{1}|Rm|^{2},$$

which is on the borderline of applying Moser iteration. Due to Bando-Kasue-Nakajima [2] (Corollary 4.10), there is an improved Kato’s inquality, namely, there are $C_{2}>0$ and $\delta\in(0,1)$, such that

$$\Delta|Rm|^{1-\delta}\leq C_{2}|Rm|^{2-\delta}.$$

Let $u=|Rm|^{1-\delta}$ and $f=|Rm|$. Then we can apply [19](Lemma 2.1) with $q=\frac{1}{1-\delta}$ and $q_{0}=\frac{1}{2(1-\delta)}$ to conclude that $|Rm|$ is in $W^{1,2}$. By Sobolev embedding we see $|Rm|\in L^{4}$. Also that $|\nabla Rm|\in L^{2}$ implies that the inequality $\Delta|Rm|\leq C_{1}|Rm|^{2}$ holds weakly on the whole ball $B$. Then we can apply the standard Moser iteration to conclude $|Rm|$ is uniformly bounded. Now consider $|\nabla Rm|$. For any $p\in B^{*}$ with $d(p,q)=r\leq 1/2$, the rescaled ball $r^{-1}B(p,r/2)$ has uniformly bounded geometry, so standard elliptic regularity for the Einstein equation then implies that $|\nabla Rm|\leq C_{3}r^{-1}.$ for some constant $C_{3}>0$. Thus $|\nabla Rm|\in L^{3}$. By Bochner formula again there is a constant $C_{4}>0$ such that

$$\Delta|\nabla Rm|\leq C_{4}|Rm||\nabla Rm|.$$

Let $u=|\nabla Rm|$, $f=C_{4}|Rm|$ and apply [19](Lemma 2.1) with $q=1$, and $q_{0}=3/4$, we get $|\nabla Rm|\in W^{1,2}$. Thus the inequality holds weakly on $B$ and by Moser iteration $|\nabla Rm|$ is uniformly bounded. Then similarly one can prove the bound for higher covariant derivatives of the curvature tensor. ∎