\begin{align*}
 & \|\partial^a \hat{h}_{\hat J_n} - \partial^a h_0 \|_{\infty} \\
 & \lesssim  (\bar J_{\max}^*(R_2))^{\frac{|a|}{d}} \bigg( (2^{-d}\bar J_{\max}^*(R_2))^{- \frac pd}  +  \tau_{2^{-d}\bar J_{\max}^*(R_2)}  \sqrt{2^{-d}\bar J_{\max}^*(R_2) \log(2^{-d}\bar J_{\max}^*(R_2)) / n}    \bigg)
\end{align*}
