Global Hypoellipticity for Systems in Time-Periodic Gelfand-Shilov Spaces

TL;DR

This paper characterizes the conditions under which overdetermined systems with time and space-dependent coefficients are globally hypoelliptic in time-periodic Gelfand-Shilov spaces, using Diophantine estimates and coefficient behavior.

Contribution

It provides necessary and sufficient conditions for global hypoellipticity of such systems, extending the understanding in the context of Gelfand-Shilov spaces.

Findings

Necessary and sufficient conditions for hypoellipticity are established.

Sign-changing behavior of coefficients influences hypoellipticity.

Normal form reduction and singular solutions characterize failure of hypoellipticity.

Abstract

We investigate the global hypoellipticity of a class of overdetermined systems with coefficients depending both on time and space variables in the setting of time-periodic Gelfand-Shilov spaces. Our main result provides necessary and sufficient conditions for the global hypoellipticity of this class of systems, stated in terms of Diophantine-type estimates and sign-changing behavior of the imaginary parts of the coefficients. Through a reduction to a normal form and detailed construction of singular solutions, we fully characterize when the system fails to be globally hypoelliptic.