= Solution
The derivative hypothesis implies that $1/Q(\xi)$ is a symbol of order $-\delta N$ at high frequency. Choose cutoffs $\chi\prec\psi$ in $X$ and a high-frequency cutoff in $\xi$. The corresponding Fourier multiplier is a <parametrix> for $Q(D)$, and the <symbol calculus>, together with the product formula from part (b), gives the localized estimate
$$
\|\chi u\|_{H^{s+\delta N}}
\leq C\left(\|\psi Q(D)u\|_{H^s}
+\|\psi u\|_{H^t}\right)
$$
for some sufficiently negative $t$. The commutator terms contain derivatives $Q^{(\alpha)}$; the assumed factor $|\xi|^{-\delta|\alpha|}$ lowers their order and lets them be absorbed inductively. Therefore
$$
\boxed{Q(D)u\in H^s_{\rm loc}(X)
\Longrightarrow u\in H^{s+\delta N}_{\rm loc}(X)}.
$$
If $Q(D)u$ is smooth, it belongs locally to $H^s$ for every $s$. Starting from the fact that every compactly supported distribution has some negative Sobolev order and repeatedly applying the gain $\delta N>0$ places $u$ in every local Sobolev space. The <Sobolev embedding theorem> then gives $u\in C^\infty(X)$. Thus $Q(D)$ is a <hypoelliptic differential operator>.
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