I am studying this book: Nonlinear Analysis on Manifolds (Emmanuel Hebey). We have Kato's inequality $$ |\nabla|\nabla^su||\leq |\nabla^{s+1}u|,\mbox{ a.e. },\forall u\in C^\infty(M), $$ and we want to proof:
Theorem 2.7: Let $(M,g)$ be a smooth, compact Riemannian $n$-manifold, $q\geq1$ real, and $m<k$ two integers. If $\frac1q<\frac{k-m}{n}$, then $H^q_k(M)\subset C^m(M)$.
In the proof, Hebey use Kato's inequality to conclude: for every $u\in C^\infty(M)$, there exist $C>0$ such that $$ \|\nabla^su\|_{H^q_{k-s}}\leq C\|u\|_{H_k^q}, $$ where $s\in\{0,\ldots,m\}$. I do not understand how to use Kato's inequality.
My try: I claim that $|\nabla^i|\nabla^su||\leq|\nabla^{i+s}u|$ a.e.. If this inequality it is true, we get $$ \|\nabla^su\|_{H^q_{k-s}}=\sum_{i=0}^{k-s}\||\nabla^i|\nabla^su||\|_{L^q}\leq\sum_{i=0}^{k-s}\||\nabla^{i+s}u|\|_{L^q}\leq\|u\|_{H^q_k}. $$ I tried proof $|\nabla^i|\nabla^su||\leq|\nabla^{i+s}u|$ by induction but I could not.