OPTIMAL GEVREY EXPONENTS FOR SOME DEGENERATE ELLIPTIC OPERATORS

We shall show first general Metivier operators ${D_y}^2+(x^{2l}+y^{2k}){D_x}^2,l,k=1,2,....,have {G_{x,y}}^{{\theta,d}}$-hypoellipticity in the vicinity of the origin (0,0), where $\theta=\frac{l(1+k)}{l(1+k)-k},\;d=\frac{\theta+k}{1+k}$ (＞1), and finally the optimality of these exponents {$\theta$, d} will be shown.