Fourier type operators on Orlicz spaces and the role of Orlicz Lebesgue exponents

Abstract

We deduce continuity and (global) wave-front properties of classes of Fourier multipliers, pseudo-differential, and Fourier integral operators when acting on Orlicz spaces, or more generally, on Orlicz-Sobolev type spaces. In particular, we extend H{\"o}rmander's improvement of Mihlin's Fourier multiplier theorem to the framework of Orlicz spaces. We also show how Young functions $\Phi$ of the Orlicz spaces are linked to properties of certain Lebesgue exponents $p_\Phi$ and $q_\Phi$ emerged from $\Phi$.