Some Properties of Solutions to Weakly Hypoelliptic Equations

Abstract EN

A linear different operator L is called weakly hypoelliptic if any local solution u of Lu=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 Lp-solution must vanish.