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 $\mathbb{T}\simeq \mathbb{R}/2\pi \mathbb{Z}$ and $Q_{j,k}(D_x)$ are pseudo-differential operators on ${\mathbb{T}}^n$. The approach consists in establishing conditions on the matrix symbol $Q(t,\xi )$ such that it can be transformed into a suitable triangular form $\Lambda (t,\xi )+\mathcal{N}(t,\xi )$, where $\Lambda (t,\xi )$ is the diagonal matrix $diag({\lambda }_1(t,\xi )\dots {\lambda }_m(t,\xi ))$ and $\mathcal{N}(t,\xi )$ is a nilpotent upper triangular matrix. Hence, the global hypoellipticity of $P$ is studied by analyzing the behavior of the eigenvalues ${\lambda }_j(t,\xi )$ and its averages ${\lambda }_{0,j}(\xi )$, as $|\xi |\to \infty$.