Hypoellipticity and vanishing theorems

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