Local Hypoellipticity by Lyapunov Function

We treat the local hypoellipticity, in the first degree, for a class of abstract differential operators complexes; the ones are given by the following differential operators: , , where is a self-adjoint linear operator, positive with , in a Hilbert space , and is a series of nonnegative powers of with coefficients in , being an open set of , for any , different from what happens in the work of Hounie (1979) who studies the problem only in the case . We provide sufficient condition to get the local hypoellipticity for that complex in the elliptic region, using a Lyapunov function and the dynamics properties of solutions of the Cauchy problem ′ , , being the first coefficient of . Besides, to get over the problem out of the elliptic region, that is, in the points ∗ such that ∗ = 0, we will use the techniques developed by Bergamasco et al. (1993) for the particular operator .