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: Lj = ∂/∂tj + (∂ϕ/∂tj)(t, A)A, j = 1,2, …, n, where A : D(A) ⊂ H → H is a self‐adjoint linear operator, positive with 0 ∈ ρ(A), in a Hilbert space H, and ϕ = ϕ(t, A) is a series of nonnegative powers of A−1 with coefficients in C∞(Ω), Ω being an open set of Rn, for any n∈N, different from what happens in the work of Hounie (1979) who studies the problem only in the case n = 1. 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 t′(s) = −∇Re ϕ0(t(s)), s ≥ 0, t(0)=t0∈Ω,ϕ0:Ω→C being the first coefficient of ϕ(t, A). Besides, to get over the problem out of the elliptic region, that is, in the points t∗ ∈Ω such that ∇Reϕ0(t∗) = 0, we will use the techniques developed by Bergamasco et al. (1993) for the particular operator A=1-Δ:H2(RN)⊂L2(RN)→L2(RN).