A remark on the stability of C∞-hypoellipticity under lower-order perturbations

We study the problem of perturbations of C∞ -hypoelliptic operators by lower order terms. We show that hypoellipticity with a finite loss of derivatives of a linear partial differential operator P (with no assumption on the characteristic set), along with its formal adjoint P∗ , is stable under perturbations by lower order linear partial differential operators whose order depends on the loss of derivatives.