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.Comment: published version (up to cosmetic issues
