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

In this paper we consider sums of squares of vector fields in R 2 \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 \[ P = X 2 + Y 2 + ∑ j = 1 L Z j 2 , P=X^2+Y^2+\sum _{j=1}^{L}Z_j^2, \] with \[ X = D x , Y = a 0 ( x , y ) x q − 1 D y , Z j = a j ( x , y ) x p j − 1 y k j D y , X=D_x, \quad Y= a_{0}(x, y) x^{q-1}{D_y}, \quad Z_j= a_{j}(x, y) x^{p_j-1}y^{k_j}\,D_y, \] with a j ( 0 , 0 ) ≠ 0 a_{j}(0, 0) \neq 0 , j = 0 , 1 , … , L j = 0, 1, \ldots , L and q > p j , { k j } q>p_j, \{k_j\} arbitrary. The theorem we prove is that P P is Gevrey-s hypoelliptic for s ≥ 1 1 − T , T = max j q − p j q k j . s\geq \frac {1}{1-T}, T = \max _j \frac {q-p_j}{q k_j}