Linear and bilinear Fourier multipliers on Orlicz modulation spaces

Abstract

AbstractLet $$\Phi _i, \Psi _i$$ Φ i , Ψ i be Young functions, $$\omega _i$$ ω i be weights and $$M^{\Phi _i,\Psi _i}_{\omega _i}(\mathbb {R} ^{d})$$ M ω i Φ i , Ψ i ( R d ) be the corresponding Orlicz modulation spaces for $$i=1,2,3$$ i = 1 , 2 , 3 . We consider linear (respect. bilinear) multipliers on $$\mathbb {R} ^{d}$$ R d , that is bounded measurable functions $$m(\xi )$$ m ( ξ ) (respect. $$m(\xi ,\eta )$$ m ( ξ , η ) ) on $$\mathbb {R} ^{d}$$ R d (respect. $$\mathbb {R} ^{2d}$$ R 2 d ) such that $$\begin{aligned} T_m(f)(x)=\int _{\mathbb {R} ^{d}}{\hat{f}}(\xi ) m(\xi )e^{2\pi i \langle \xi , x\rangle }d\xi \end{aligned}$$ T m ( f ) ( x ) = ∫ R d f ^ ( ξ ) m ( ξ ) e 2 π i ⟨ ξ , x ⟩ d ξ (respect. $$\begin{aligned} B_m(f_1,f_2)(x)=\int _{\mathbb {R} ^{d}}\int _{\mathbb {R} ^{d}} \hat{f_1}(\xi ) \hat{f_2}(\eta )m(\xi ,\eta )e^{2\pi i \langle \xi +\eta , x\rangle }d\xi d\eta \end{aligned}$$ B m ( f 1 , f 2 ) ( x ) = ∫ R d ∫ R d f 1 ^ ( ξ ) f 2 ^ ( η ) m ( ξ , η ) e 2 π i ⟨ ξ + η , x ⟩ d ξ d η define a bounded linear (respect. bilinear) operator from $$M^{\Phi _1,\Psi _1}_{\omega _1}(\mathbb {R} ^{d})$$ M ω 1 Φ 1 , Ψ 1 ( R d ) to $$M^{\Phi _2,\Psi _2}_{\omega _2}(\mathbb {R} ^{d})$$ M ω 2 Φ 2 , Ψ 2 ( R d ) (respect. $$M^{\Phi _1,\Psi _1}_{\omega _1}(\mathbb {R} ^{d})\times M^{\Phi _2,\Psi _2}_{\omega _2}(\mathbb {R} ^{d})$$ M ω 1 Φ 1 , Ψ 1 ( R d ) × M ω 2 Φ 2 , Ψ 2 ( R d ) to $$M^{\Phi _3,\Psi _3}_{\omega _3}(\mathbb {R} ^{d})$$ M ω 3 Φ 3 , Ψ 3 ( 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.