Fourier Type Operators on Orlicz Spaces and the Role of Orlicz Lebesgue Exponents

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 Hörmander’s improvement of Mihlin’s Fourier multiplier theorem to the framework of Orlicz spaces. We also show how Young functions \(\Phi \) $\mathrm{\Phi }$ of the Orlicz spaces are linked to properties of certain Lebesgue exponents \(p_\Phi \) $p_{\mathrm{\Phi }}$ and \(q_\Phi \) $q_{\mathrm{\Phi }}$ emerged from \(\Phi \) $\mathrm{\Phi }$ .