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

TL;DR

This paper investigates the conditions under which a class of pseudo-differential operators on the torus are globally hypoelliptic, revealing number-theoretical obstructions and the influence of symbol growth and coefficient behavior.

Contribution

It establishes necessary and sufficient conditions for global hypoellipticity of these operators, including the role of symbol growth, coefficient interactions, and irrational approximations.

Findings

Number-theoretical obstructions are necessary for hypoellipticity.

Logarithmic growth of the symbol ensures sufficiency of conditions.

Super-logarithmic growth requires analyzing sign changes and coefficient interactions.

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.