On the Stability of the $ C^\infty$-Hypoellipticity under Perturbations

We study the problem of perturbations of $ C^\infty $-hypoelliptic operators by lower order terms. After giving several examples which show many different possibilities, we then prove a stability result which shows that a hypoelliptic linear partial differential operator $ P$ which loses finitely many derivatives and whose formal adjoint $ P^*$ is still hypoelliptic (but with no assumption on the loss of derivatives) remains hypoelliptic with the same loss of derivatives after perturbation by a lower order linear partial differential operator (whose order depends on the loss of derivatives).