Global hypoellipticity for a class of pseudo-differential operators on the torus

Abstract

We show that an obstruction of number-theoretical nature appears as a necessary condition for the global hypoellipticity of the pseudo-differential operator $L=D_t+(a+ib)(t)P(D_x)$ on $\mathbb{T}^1_t\times\mathbb{T}_x^{N}$. This condition is also sufficient when the symbol $p(\xi)$ of $P(D_x)$ has at most logarithmic growth. If $p(\xi)$ has super-logarithmic growth, we show that the global hypoellipticity of $L$ depends on the change of sign of certain interactions of the coefficients with the symbol $p(\xi).$ Moreover, the interplay between the order of vanishing of coefficients with the order of growth of $p(\xi)$ plays a crucial role in the global hypoellipticity of $L$. We also describe completely the global hypoellipticity of $L$ in the case where $P(D_x)$ is positively homogeneous. Additionally, we explore the influence of irrational approximations of a real number in the global hypoellipticity.