On the global properties of Fourier multipliers in the nonharmonic analysis setting

TL;DR

This paper studies Fourier multipliers in nonharmonic analysis, providing conditions for global hypoellipticity and solvability, and extends known results to non-periodic boundary conditions.

Contribution

It offers new necessary and sufficient conditions for Fourier multipliers to be globally hypoelliptic and solvable in nonharmonic boundary value problems.

Findings

Established criteria for global hypoellipticity of Fourier multipliers.

Derived conditions for global solvability in nonharmonic settings.

Extended periodic case results to non-periodic boundary conditions.

Abstract

In this paper, we investigate the global properties of Fourier multipliers in the setting of nonharmonic analysis of boundary value problems. We give necessary and sufficient conditions for a Fourier multiplier to be globally hypoelliptic and also to be globally solvable. As an application, we consider operators on $[0,1]^2$ with non-periodic boundary conditions and we obtain results that extend what is already known in the periodic case.