Gevrey hypoellipticity for sums of squares with a non-homogeneous degeneracy

Abstract:

In this paper we consider sums of squares of vector fields in $\mathbb {R}^2$ satisfying Hörmander’s condition and with polynomial, but non-(quasi-)homoge- neous, coefficients. We obtain a Gevrey hypoellipticity index which we believe to be sharp. The general operator we consider is \[ P=X^2+Y^2+\sum _{j=1}^{L}Z_j^2, \] with \[ X=D_x, \quad Y= a_{0}(x, y) x^{q-1}{D_y}, \quad Z_j= a_{j}(x, y) x^{p_j-1}y^{k_j} D_y, \] with $a_{j}(0, 0) \neq 0$, $j = 0, 1, \ldots , L$ and $q>p_j, \{k_j\}$ arbitrary. The theorem we prove is that $P$ is Gevrey-s hypoelliptic for $s\geq \frac {1}{1-T}, T = \max _j \frac {q-p_j}{q k_j}.$