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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 estimatefor 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

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.

The heat operatoris hypoelliptic: its symbol $i\tau +|\xi {|}^2$ satisfies the derivative estimates that yield local regularity by the argument in part (c). It is not elliptic as an operator of total order two, because its principal symbol is $|\xi {|}^2$, which vanishes at every nonzero covector $(\tau ,0)$ with $\tau \ne 0$. Hence it is a hypoelliptic differential operator that is not an elliptic differential operator.