Abstract
A linear different operatorLis called weakly hypoelliptic if any local solutionuofLu=0is 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 anyLp-solution must vanish.
Highlights
Hypoelliptic partial differential equations form a huge class of linear PDEs many of which are very important in applications
We show by example that the class of weakly hypoelliptic operators is strictly larger than that of hypoelliptic operators
The class of weakly hypoelliptic operators allows for a certain degeneracy of the principal symbol on “small sets” and might be of interest for geometric applications
Summary
Hypoelliptic partial differential equations form a huge class of linear PDEs many of which are very important in applications. The study of hypoelliptic operators was initiated by Hormander and others; see, for example, [1–4] We generalize this class of operators even further by only demanding that any solution u to Lu = 0 be smooth. We show that even a slightly stronger version of the Montel theorem holds for solutions to any weakly hypoelliptic equation: any locally L1-bounded sequence subconverges in the C∞-topology to a solution (Theorem 4). In case the underlying domain is Rn and the weakly hypoelliptic operator has constant coefficients and satisfies a weighed homogeneity condition, we show that the Liouville theorem holds: any bounded solution must be constant (Theorem 7), and any Lp-solution must be zero (Theorem 12). As we will see, the proofs of the general statements are rather simple
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