Time-periodic Gelfand-Shilov spaces and global hypoellipticity on $\mathbb{T} \times \mathbb{R}^n$

TL;DR

This paper introduces time-periodic Gelfand-Shilov spaces on a torus times Euclidean space, develops Fourier analysis for these spaces, and characterizes the global hypoellipticity of certain linear differential operators.

Contribution

It defines new function spaces combining periodic and Gelfand-Shilov properties, and provides a framework to analyze hypoellipticity of evolution operators in this setting.

Findings

Characterization of time-periodic Gelfand-Shilov spaces.

Fourier analysis based on eigenfunction expansions.

Criteria for global hypoellipticity of differential operators.

Abstract

We introduce a class of time-periodic Gelfand-Shilov spaces of functions on $\mathbb{T} \times \mathbb{R}^n$, where $\mathbb{T} \sim \mathbb{R} /2\pi \mathbb{Z}$ is the one-dimensional torus. We develop a Fourier analysis inspired by the characterization of the Gelfand-Shilov spaces in terms of the eigenfunction expansions given by a fixed normal, globally elliptic differential operator on $\mathbb{R}^n$. In this setting, as an application, we characterize the global hypoellipticity for a class of linear differential evolution operators on $\mathbb{T} \times \mathbb{R}^n$.