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, whereA:D(A)⊂H→His a self-adjoint linear operator, positive with0∈ρ(A), in a Hilbert spaceH, andϕ=ϕ(t,A)is a series of nonnegative powers ofA-1with coefficients inC∞(Ω),Ωbeing an open set ofRn, for anyn∈N, different from what happens in the work of Hounie (1979) who studies the problem only in the casen=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 problemt′(s)=-∇Reϕ0(t(s)),s≥0,t(0)=t0∈Ω,ϕ0:Ω→Cbeing the first coefficient ofϕ(t,A). Besides, to get over the problem out of the elliptic region, that is, in the pointst∗ ∈Ωsuch that∇Reϕ0(t∗)= 0, we will use the techniques developed by Bergamasco et al. (1993) for the particular operatorA=1-Δ:H2(RN)⊂L2(RN)→L2(RN).