Notes on bilinear multipliers on Orlicz spaces
Let $Φ_1 , Φ_2 $ and $ Φ_3$ be Young functions and let $L^{Φ_1}(\mathbb{R})$, $L^{Φ_2}(\mathbb{R})$ and $L^{Φ_3}(\mathbb{R})$ be the corresponding Orlicz spaces. We say that a function $m(ξ,η)$ defined on $\mathbb{R}\times \mathbb{R}$ is a bilinear multiplier of type $(Φ_1,Φ_2,Φ_3)$ if \[ B_m(f,g)(x)=\int_\mathbb{R} \int_\mathbb{R} \hat{f}(ξ) \hat{g}(η)m(ξ,η)e^{2πi (ξ+η) x}dξdη\] defines a bounded bilinear operator from $L^{Φ_1}(\mathbb{R}) \times L^{Φ_2}(\mathbb{R})$ to $L^{Φ_3}(\mathbb{R})$. We denote by $BM_{(Φ_1,Φ_2,Φ_3)}(\mathbb{R})$ the space of all bilinear multipliers of type $(Φ_1,Φ_2,Φ_3)$ and investigate some properties of such a class. Under some conditions on the triple $(Φ_1,Φ_2,Φ_3)$ we give some examples of bilinear multipliers of type $(Φ_1,Φ_2,Φ_3)$. We will focus on the case $m(ξ,η)=M(ξ-η) $ and get necessary conditions on $(Φ_1,Φ_2,Φ_3)$ to get non-trivial multipliers in this class. In particular we recover some of the the known results for Lebesgue spaces.