Solution
= Solution
The heat operator
$$
\partial_t-\Delta_x
$$
is 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 $\tau\ne0$. Hence it is a <hypoelliptic differential operator> that is not an <elliptic differential operator>.