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

We find that if a Fourier multiplier is continuous from $L^{{\mathrm{\Phi }}_1}$ to $L^{{\mathrm{\Phi }}_2}$, then it is also continuous from $M^{{\mathrm{\Phi }}_1,\mathrm{\Psi }}$ to $M^{{\mathrm{\Phi }}_2,\mathrm{\Psi }}$, where ${\mathrm{\Phi }}_1,{\mathrm{\Phi }}_2,\mathrm{\Psi }$ are quasi-Young functions and ${\mathrm{\Phi }}_1$ fulfills the ${\mathrm{\Delta }}_2$-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 $r\in (d\mathrm{/}(d+2),1]$ and $\alpha \in (d(1-r)\mathrm{/}r,2]$.