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

TL;DR

This paper extends classical Fourier multiplier theorems to quasi-Banach Orlicz and Orlicz modulation spaces, establishing boundedness results under new conditions and broadening the scope of harmonic analysis tools.

Contribution

It demonstrates that Fourier multipliers continuous on Orlicz spaces are also continuous on associated Orlicz modulation spaces, extending key theorems like Mihlin's and H"ormander's to these settings.

Findings

Fourier multipliers are bounded on certain Orlicz modulation spaces.

Mihlin's and H"ormander's theorems hold in Orlicz modulation spaces.

Boundedness of specific Fourier multipliers with phase functions on quasi-Banach 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 $\alpha\in\big(d(1-r)/r, 2\big]$.