Global Hypoellipticity for a Class of Pseudo-differential Operators on the Torus

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_ttimes 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 homogeneous. Additionally, we explore the influence of irrational approximations of a real number in the global hypoellipticity.