Linear and bilinear Fourier multipliers on Orlicz modulation spaces

Abstract

Let \(\Phi _i, \Psi _i\) be Young functions, \(\omega _i\) be weights and \(M^{\Phi _i,\Psi _i}_{\omega _i}(\mathbb {R} ^{d})\) be the corresponding Orlicz modulation spaces for \(i=1,2,3\). We consider linear (respect. bilinear) multipliers on \(\mathbb {R} ^{d}\), that is bounded measurable functions \(m(\xi )\) (respect. \(m(\xi ,\eta )\)) on \(\mathbb {R} ^{d}\) (respect. \(\mathbb {R} ^{2d}\)) such that

$$\begin{aligned} T_m(f)(x)=\int _{\mathbb {R} ^{d}}{\hat{f}}(\xi ) m(\xi )e^{2\pi i \langle \xi , x\rangle }d\xi \end{aligned}$$

(respect.

$$\begin{aligned} B_m(f_1,f_2)(x)=\int _{\mathbb {R} ^{d}}\int _{\mathbb {R} ^{d}} \hat{f_1}(\xi ) \hat{f_2}(\eta )m(\xi ,\eta )e^{2\pi i \langle \xi +\eta , x\rangle }d\xi d\eta \end{aligned}$$

define a bounded linear (respect. bilinear) operator from \(M^{\Phi _1,\Psi _1}_{\omega _1}(\mathbb {R} ^{d})\) to \(M^{\Phi _2,\Psi _2}_{\omega _2}(\mathbb {R} ^{d})\) (respect. \(M^{\Phi _1,\Psi _1}_{\omega _1}(\mathbb {R} ^{d})\times M^{\Phi _2,\Psi _2}_{\omega _2}(\mathbb {R} ^{d})\) to \(M^{\Phi _3,\Psi _3}_{\omega _3}(\mathbb {R} ^{d})\)). In this paper we study some properties of these spaces and give methods to generate linear and bilinear multipliers between Orlicz modulation spaces.