 Solution

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.