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 $H^{l-k}(M)$?
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$\begingroup$ I haven't tried to write the details down, but it appears to me that you should be able to prove this using just the definition of the norm on $H^k$, the definition of a dual (operator) norm, and maybe integration by parts. $\endgroup$– Deane YangCommented Mar 8, 2012 at 18:54
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If $P$ is a degree $ l $ differential operator, then
$$ \Vert Pu \Vert_s < C \Vert u \Vert_{s+l} ,\;\; \forall u, s. $$
Now take $s=-k$, $P=\nabla^l$.
answered Mar 8, 2012 at 19:01