A remark on the stability of $$C^\infty $$ C ∞ -hypoellipticity under lower-order perturbations

Abstract

We study the problem of perturbations of $$C^\infty $$ -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.