Globally hypoelliptic triangularizable systems of periodic pseudo-differential operators

Abstract

This article presents an investigation on the global hypoellipticity problem for systems belonging to the class $P = D_t + Q(t,D_x)$, where $Q(t,D_x)$ is a $m\times m$ matrix with entries $c_{j,k}(t)Q_{j,k}(D_x)$. The coefficients $c_{j,k}(t)$ are smooth, complex-valued functions on the torus $\mathbb{T} \simeq \mathbb{R}/2π\mathbb{Z}$ and $Q_{j,k}(D_x)$ are pseudo-differential operators on $ \mathbb{T}^n$. The approach consists in establishing conditions on the matrix symbol $Q(t,ξ)$ such that it can be transformed into a suitable triangular form $Λ(t,ξ) + \mathcal{N}(t,ξ)$, where $Λ(t,ξ)$ is the diagonal matrix $diag(λ_{1}(t,ξ) \ldots λ_{m}(t,ξ))$ and $\mathcal{N}(t,ξ)$ is a nilpotent upper triangular matrix. Hence, the global hypoellipticity of $P$ is studied by analyzing the behavior of the eigenvalues $λ_{j}(t,ξ)$ and its averages $λ_{0,j}(ξ)$, as $|ξ| \to \infty$.