Local Hypoellipticity by Lyapunov Function

Abstract

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)=-∇Reϕ0(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).