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

TL;DR

This paper investigates the conditions under which overdetermined systems of pseudo-differential operators on the torus are globally hypoelliptic, linking it to their normal forms and number-theoretical obstructions.

Contribution

It establishes necessary and sufficient conditions for global hypoellipticity of these systems, including the role of number-theoretical obstructions and growth hypotheses.

Findings

Number-theoretical obstructions are necessary for hypoellipticity.

Normal form analysis links hypoellipticity to simpler systems.

Growth conditions influence sufficiency of hypoellipticity.

Abstract

This article studies the global hypoellipticity of a class of overdetermined systems of pseudo-differential operators defined on the torus. The main goal consists in establishing connections between the global hypoellipticity of the system and the global hypoellipticity of its normal form. It is proved that an obstruction of number-theoretical nature appears as a necessary condition to the global hypoellipticity. Conversely, the sufficiency is approached ana\-lyzing three types of hypotheses: a H\"{o}rmander condition, logarithmic growth and super-logarithmic growth.