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 H\"ormander'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 $r\in\big(d/(d+2),1\big]$ and $\alpha\in\big(d(1-r)/r, 2\big]$.