Some Properties of Solutions to Weakly Hypoelliptic Equations

Abstract

A linear different operator $L$ is called weakly hypoelliptic if any local solution $u$ of $Lu=\mathrm{0}$ is smooth. We allow for systems, that is, the coefficients may be matrices, not necessarily of square size. This is a huge class of important operators which coverall elliptic, overdetermined elliptic, subelliptic, and parabolic equations. We extend several classical theorems from complex analysis to solutions of any weakly hypoelliptic equation: the Montel theorem providing convergent subsequences, the Vitali theorem ensuring convergence of a given sequence, and Riemann's first removable singularity theorem. In the case of constant coefficients, we show that Liouville's theorem holds, any bounded solution must be constant, and any $L^p$-solution must vanish.