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 ${\mathbb{T}}_t^1\times {\mathbb{T}}_x^N$. 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.

37 pages; updated version

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