Fourier Multipliers on Quasi-Banach Orlicz Spaces and Orlicz Modulation Spaces

Abstract

We find that if a Fourier multiplier is continuous from $L^{{\Phi }_1}$ to $L^{{\Phi }_2}$, then it is also continuous from $M^{{\Phi }_1,\Psi }$ to $M^{{\Phi }_2,\Psi }$, where ${\Phi }_1,{\Phi }_2,\Psi$ are quasi-Young functions and ${\Phi }_1$ fulfills the ${\Delta }_2$-condition. This result is applied to show that Mihlin's Fourier multiplier theorem and Hörmander's improvement hold in certain Orlicz modulation spaces. Lastly, we show that the Fourier multiplier with symbol $m(\xi )=e^{i\mu (\xi )}$, where $\mu$ is homogeneous of order $\alpha$, is bounded on quasi-Banach Orlicz modulation spaces of order $r$, assuming $r\in (d/(d+2),1]$ and $\alpha \in (d(1-r)/r,2]$.

15 pages. This is the fifth version