Linear and bilinear Fourier multipliers on Orlicz modulation spaces

Let \(\Phi _i, \Psi _i\) ${\mathrm{\Phi }}_i,{\mathrm{\Psi }}_i$ be Young functions, \(\omega _i\) ${\omega }_i$ be weights and \(M^{\Phi _i,\Psi _i}_{\omega _i}(\mathbb {R} ^{d})\) $M_{{\omega }_i}^{{\mathrm{\Phi }}_i,{\mathrm{\Psi }}_i}({\mathbb{R}}^d)$ be the corresponding Orlicz modulation spaces for \(i=1,2,3\) $i=1,2,3$ . We consider linear (respect. bilinear) multipliers on \(\mathbb {R} ^{d}\) ${\mathbb{R}}^d$ , that is bounded measurable functions \(m(\xi )\) $m(\xi )$ (respect. \(m(\xi ,\eta )\) $m(\xi ,\eta )$ ) on \(\mathbb {R} ^{d}\) ${\mathbb{R}}^d$ (respect. \(\mathbb {R} ^{2d}\) ${\mathbb{R}}^{2d}$ ) such that \(\begin{aligned} T_m(f)(x)=\int _{\mathbb {R} ^{d}}{\hat{f}}(\xi ) m(\xi )e^{2\pi i \langle \xi , x\rangle }d\xi \end{aligned}\) $$\begin{array}{c}\hfill T_m(f)(x)={\int }_{{\mathbb{R}}^d}\widehat{f}(\xi )m(\xi )e^{2\pi i⟨\xi ,x⟩}d\xi \end{array}$$ (respect. \(\begin{aligned} B_m(f_1,f_2)(x)=\int _{\mathbb {R} ^{d}}\int _{\mathbb {R} ^{d}} \hat{f_1}(\xi ) \hat{f_2}(\eta )m(\xi ,\eta )e^{2\pi i \langle \xi +\eta , x\rangle }d\xi d\eta \end{aligned}\) $$\begin{array}{c}\hfill B_m(f_1,f_2)(x)={\int }_{{\mathbb{R}}^d}{\int }_{{\mathbb{R}}^d}\widehat{f_1}(\xi )\widehat{f_2}(\eta )m(\xi ,\eta )e^{2\pi i⟨\xi +\eta ,x⟩}d\xi d\eta \end{array}$$ define a bounded linear (respect. bilinear) operator from \(M^{\Phi _1,\Psi _1}_{\omega _1}(\mathbb {R} ^{d})\) $M_{{\omega }_1}^{{\mathrm{\Phi }}_1,{\mathrm{\Psi }}_1}({\mathbb{R}}^d)$ to \(M^{\Phi _2,\Psi _2}_{\omega _2}(\mathbb {R} ^{d})\) $M_{{\omega }_2}^{{\mathrm{\Phi }}_2,{\mathrm{\Psi }}_2}({\mathbb{R}}^d)$ (respect. \(M^{\Phi _1,\Psi _1}_{\omega _1}(\mathbb {R} ^{d})\times M^{\Phi _2,\Psi _2}_{\omega _2}(\mathbb {R} ^{d})\) $M_{{\omega }_1}^{{\mathrm{\Phi }}_1,{\mathrm{\Psi }}_1}({\mathbb{R}}^d)\times M_{{\omega }_2}^{{\mathrm{\Phi }}_2,{\mathrm{\Psi }}_2}({\mathbb{R}}^d)$ to \(M^{\Phi _3,\Psi _3}_{\omega _3}(\mathbb {R} ^{d})\) $M_{{\omega }_3}^{{\mathrm{\Phi }}_3,{\mathrm{\Psi }}_3}({\mathbb{R}}^d)$ ). In this paper we study some properties of these spaces and give methods to generate linear and bilinear multipliers between Orlicz modulation spaces.