Subelliptic estimates for some systems of complex vector fields: Quasihomogeneous case

Abstract

For about twenty five years it was a kind of folk theorem that complex vector-fields defined on Ω × R t \Omega \times \mathbb R_t (with Ω \Omega open set in R n \mathbb R^n ) by \[ L j = ∂ ∂ t j + i ∂ φ ∂ t j ( t ) ∂ ∂ x , j = 1 , … , n , t ∈ Ω , x ∈ R , L_j = \frac {\partial }{\partial t_j} + i \frac {\partial \varphi }{\partial t_j}(\mathbf {t})\, \frac {\partial }{\partial x}\;,\; j=1,\dots , n\;,\; \mathbf {t}\in \Omega , x\in \mathbb R, \] with φ \varphi analytic, were subelliptic as soon as they were hypoelliptic. This was the case when n = 1 n=1 , but in the case n > 1 n>1 , an inaccurate reading of the proof given by Maire (see also Trèves) of the hypoellipticity of such systems, under the condition that φ \varphi does not admit any local maximum or minimum (through a nonstandard subelliptic estimate), was supporting the belief for this folk theorem. Quite recently, J.L. Journé and J.M. Trépreau show by examples that there are very simple systems (with polynomial φ \varphi ’s) which are hypoelliptic but not subelliptic in the standard L 2 L^2 -sense. So it is natural to analyze this problem of subellipticity which is in some sense intermediate (at least when φ \varphi is C ∞ C^\infty ) between the maximal hypoellipticity (which was analyzed by Helffer-Nourrigat and Nourrigat) and the simple local hypoellipticity (or local microhypoellipticity) and to start first with the easiest nontrivial examples. The analysis presented here is a continuation of a previous work by the first author and is devoted to the case of quasihomogeneous functions.