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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}.

We investigate the classical solvability for some classes of linear, degenerate equations in divergence form with prescribed Dirichlet data. Since the boundary value problem is characteristic according to Fichera on a part of the boundary, some typical nonlinear phenomena at these points are observed as boundary gradient blowups of the classical solutions in space directions. The regularity results explain the lack of hypoellipticity for special right-hand sides or boundary data for linear degenerate parabolic equations.