A pseudodifferential calculus for maximally hypoelliptic operators and the Helffer-Nourrigat conjecture

We extend the classical regularity theorem of elliptic operators to maximally hypoelliptic differential operators. More precisely, given vector fields $X_1,\dots ,X_m$ on a smooth manifold which satisfy H"ormander’s bracket generating condition, we define a principal symbol for $\mathit{\text{any}}$ linear differential operator. Our symbol takes into account the vector fields $X_i$ and their commutators. We show that for an arbitrary differential operator, its principal symbol is invertible if and only if the operator is maximally hypoelliptic. This answers affirmatively a conjecture due to Helffer and Nourrigat. Our result is proven in a more general setting, where we allow each one of the vector fields $X_1,\dots ,X_m$ to have an arbitrary weight. In particular, our theorem generalizes H"ormander’s sum of squares theorem to higher order polynomials.