Analytic Hypoellipticity for a New Class of Sums of Squares of Vector Fields in $${\mathcal {R}}^3$$ R 3

Abstract

This paper is devoted to a substantial generalization of previous work on the analytic hypoellipticity of sums of squares \(P=\sum _1^4X^2_j\) of real vector fields with real analytic coefficient in three variables. For p(x, y) quasi-homogeneous in (x, y), consider the vector fields $$\begin{aligned} X_1 = \frac{\partial }{\partial x}, \quad X_2=-\frac{\partial }{\partial y} + p(x,y)\frac{\partial }{\partial t}, \quad X_3=x^{n_1}\frac{\partial }{\partial t}, \quad X_4=y^{n_2}\frac{\partial }{\partial t}, \end{aligned}$$ \( n_1, n_2 \ne 0\). We show that the operator $$\begin{aligned} P=\sum _1^4 X_j^2, \end{aligned}$$ well known to be \(C^\infty \)-hypoelliptic, is actually analytic hypoelliptic near the origin in \({\mathcal {R}}^3\).