On the Subellipticity of Some Hypoelliptic Quasihomogeneous Systems of Complex Vector Fields

Abstract

For about twenty five years it was a kind of folk theorem that complex vector-fields defined on Ω × ℝ t (with Ω open set in ℝ n ) by $$ L_j = \frac{\partial } {{\partial t_j }} + i\frac{{\partial \phi }} {{\partial t_j }}(t)\frac{\partial } {{\partial x}},j = 1, \ldots ,n, t \in \Omega ,x \in \mathbb{R}, $$ with φ analytic, were subelliptic as soon as they were hypoelliptic. This was indeed the case when n = 1 [Tr1] but in the case n > 1, an inaccurate reading of the proof (based on a non standard subelliptic estimate) given by Maire [Mai1] (see also Treves [Tr2]) of the hypoellipticity of such systems, under the condition that φ does not admit any local maximum or minimum, was supporting the belief for this folk theorem. This question reappears in the book of [HeNi] in connection with the semi-classical analysis of Witten Laplacians. Quite recently, J.L. Journe and J.M. Trepreau [JoTre] show by explicit examples that there are very simple systems (with polynomial φ’s) which were hypoelliptic but not subelliptic in the standard L 2-sense. But these operators are not quasihomogeneous.