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

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.