Under the assumption of Proposition 6.1, for any $0<\theta<1$, we have $||D^{k}\phi||_{0,B_{\theta}}\leq C(k)$, for any $k\geq 2$. Here $C(k)$ has the same dependence as described in Proposition 6.1 besides dependence on $\theta$ and on $||\phi||_{0,B_{1}}$.