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.