Hypoellipticity and Vanishing Theorems

Let −iLT (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 ≥ 1, the latter formally normal, both acting on sections of a vector bundle over a closed manifold. It is shown that if P + (−iLT ) m is elliptic then the restriction of −iLT to D ⊂ kerP ⊂ L (D is carefully specified) yields a selfadjoint operator −iLT |D : D ⊂ kerP → kerP with compact resolvent. It is also shown that, in the presence of an additional hypothesis on microlocal hypoellipticity of P , −iLT |D 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.