Let $(M,g)$ be a closed, compact Riemannian manifold. Let $u$ be a smooth function. Let $H^{-k}(M)$,, $k$ is a positive integer, be the dual Hilbert space of $H^{k}(M)$. Does it follow that $|| |\nabla^l u| ||^{-k}$, where $||\circ||^{-k}$ is the norm on $H^{-k}(M)$ and $l \in \Bbb{Z}\_+$, can be bounded by $|| u ||^{l-k}$, where $||\circ||^{l-k}$ is the norm on $H^{l-k}(M)$? 



Estimating norms of derivatives
If $P$ is a degree $ l $ differential operator, then


$$ \Vert Pu \Vert\_s < C \Vert u \Vert\_{s+l} ,\;\; \forall u, s. $$


Now take $s=-k$, $P=\nabla^l$. 


