Hypoellipticity and vanishing theorems

Let $-\im\Lie_\T$ (essentially Lie derivative with respect to $\T$, a smooth nowhere zero real vector field) and $P$ be commuting differential operators, respectively of orders 1 and $m\geq 1$, the latter formally normal, both acting on sections of a vector bundle over a closed manifold. It is shown that if $P+(-i\Lie_\T)^m$ is elliptic then the restriction of $-\im\Lie_\T$ to $\Dom\subset \ker P\subset L^2$ yields a selfadjoint operator $-\im\Lie_\T|_\Dom:\Dom\subset\ker P\to \ker P$ with compact resolvent ($\Dom$ is specified carefully). It is also shown that, in the presence of an additional hypothesis on microlocal hypoellipticity of $P$, $-\im\Lie_\T|_\Dom$ is semi-bounded. These results are applied to CR manifolds on which $\T$ acts as an infinitesimal CR transformation which are then shown to yield versions of Kodaira's vanishing theorem.