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 ℝⁿ) 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 Trèves [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. Journé and J.M. Trépreau [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.

In [De] and [DeHe] the homogeneous and the quasihomogeneous cases were analyzed in dimension 2. Large classes of systems for which subellipticiity can be proved were exhibited. We will show in this paper how a new idea for the construction of escaping rays permits to show that in the analytic case all the quasihomogeneous hypoelliptic systems in the class above considered by Maire are effectively subelliptic in the 2-dimensional case. The analysis presented here is a continuation of two previous works by the first author for the homogeneous case [De] and the two authors for the quasihomogeneous case [DeHe].

In honor of Linda P. Rothschild