A characterization of hypoelliptic differential operators with variable coefficients

Abstract:

Let $P$ be a linear differential operator with coefficients in ${C^\infty }(\Omega )$ where $\Omega \subset {{\mathbf {R}}^n}$. We characterize the hypoelliptic operators in terms of the $\ast$-hypoelliptic operators. $P$ is defined to be $\ast$-hypoelliptic on $\Omega$ if and only if $u \in {\mathcal {D}’_F}(\Omega )$ and $Pu \in {C^\infty }(\Omega )$ imply $u \in {C^\infty }(\Omega )$. We characterize the $\ast$-hypoelliptic operators via a priori estimates. We prove $P$ is hypoelliptic on $\Omega$ if and only if for $u \in \mathcal {D}’(\Omega )$ and $Pu \in {C^\infty }(\Omega ’)$ with $\Omega ’ \subset \Omega$, there exists for each ${x_0} \in \Omega ’$ a relatively compact open neighborhood ${\Omega _{{x_0}}} \subset \Omega ’$ of ${x_0}$ such that $P$ is $\ast$-hypoelliptic on ${\Omega _{{x_0}}}$.