Some properties of solutions to weakly hypoelliptic equations

TL;DR

This paper studies weakly hypoelliptic equations, a broad class including elliptic and parabolic types, extending classical complex analysis theorems to their solutions and establishing properties like Liouville's theorem.

Contribution

It generalizes classical theorems such as Montel, Vitali, and Riemann's removable singularity to solutions of weakly hypoelliptic equations, including systems with matrix coefficients.

Findings

Classical theorems extended to weakly hypoelliptic equations

Liouville's theorem holds for constant coefficient cases

Bounded solutions are constant, L^p solutions vanish

Abstract

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 cover all 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.