Complex sum of squares of vector fields (hypoelliptic operators)

Question

Consider a compact $M$ of dimension $n$. Consider real smooth vector fields $X_0, X_1, X_2,..., X_n$ on $M$ and consider the differential operator $\mathcal{L}_1 = \sum_{i = 1}^n X_i^2 + X_0.$ Now, by a result of Rothschild and Stein (see here, Theorems 16 and 18), if the commutators of weight $\leq r$ of the above vector fields generate $T_x(M)$ for all $x \in M$, then $\mathcal{L}_1$ is hypoelliptic with loss of $\frac{2}{r}$ derivatives.

My question is, is something known for vector fields with complex coefficients? For example, consider the differential operator $\mathcal{L}_2 = \sum_{i = 1}^n X_i^2 + i X_0,$ where the commutators of weight $\leq r$ of the above vector fields generate $T_x(M)$ for all $x \in M$. Is it known what the loss is in this case? If a general result is not known, I would also highly appreciate references where specific examples are computed. Thanks in advance!

Answer 1

While this question is old, it's perhaps worth noting that something is known: It's known that the situation can be very complicated.

For example, Kohn (Annals of Mathematics, 162 (2005), 943–986) showed that the $L^2$ sum of squares of complex vector fields satisfying Hormander's condition can be hypoelliptic while failing to be subelliptic: it can be hypoelliptic with as loss of as many derivatives as you like.

Based on Kohn's example, Christ showed that the $L^2$ sum of squares of complex vector fields satisfying Hormander's condition can fail to be hypoelliptic.

While there have been several papers extending the above ideas, I don't know of any general theory. The situation is currently more opaque than the case of real vector fields.