Assume $\Phi$ is some diffeomorphism of a certain manifold. Let $\Phi^{-1}$ denote the inverse map and let $(D\Phi)^{-1}$ denote the matrix inverse of $D\Phi$.

>**QUESTION.** Does this norm estimate hold? (The sum is over all $n$-tuples of non-negative integers.)
$$\Vert D^n(\Phi^{-1})\Vert \lesssim \Vert (D\Phi)^{-1}\Vert^{2n-1}\cdot 
\sum_{\substack{s_1+\cdots+s_n=n-1 \\ s_1+2s_2+\cdots+ns_n=2n-2}}
\Vert D\Phi\Vert^{s_1} \Vert D^2\Phi\Vert^{s_2}\cdots \Vert D^n\Phi\Vert^{s_n}.$$

It is asking for the multivariable version of [this MO question][1].

[1]: https://mathoverflow.net/questions/98501/fa%c3%a0-di-brunos-formula-for-inverse-functions