Schwartz regularity of differential operators on the cylinder

TL;DR

This paper investigates the Schwartz regularity and solvability of differential operators on the cylinder using mixed Fourier analysis, providing necessary and sufficient conditions for their hypoellipticity and solvability.

Contribution

It introduces a novel approach employing mixed Fourier transforms to characterize Schwartz global hypoellipticity and solvability of differential operators on the cylinder.

Findings

Derived necessary and sufficient conditions for Schwartz global hypoellipticity.

Established criteria for Schwartz global solvability.

Utilized mixed Fourier analysis to analyze operator properties.

Abstract

This article presents an investigation of global properties of a class of differential operators on $\mathbb{T}^1\times\mathbb{R}$. Our approach involves the utilization of a mixed Fourier transform, incorporating both partial Fourier series on the torus and partial Fourier transform in Euclidean space. By examining the behaviour of the mixed Fourier coefficients, we obtain necessary and sufficient conditions for the Schwartz global hypoellipticity of this class of differential operators, as well as conditions for the Schwartz global solvability of said operators.