Local Hypoellipticity by Lyapunov Function

الملخص EN

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: L j = ∂ / ∂ t j + ( ∂ ϕ / ∂ t j ) ( 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 R n , 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 ) = - ∇ R e ϕ 0 ( t ( s ) ) , s ≥ 0 , t ( 0 ) = t 0 ∈ Ω , ϕ 0 : 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 ∇ R e ϕ 0 ( t ∗ ) = 0, we will use the techniques developed by Bergamasco et al.

(1993) for the particular operator A = 1 - Δ : H 2 ( R N ) ⊂ L 2 ( R N ) → L 2 ( R N ) .