Notes on bilinear multipliers on Orlicz spaces

Abstract

Let $\Phi_1 , \Phi_2 $ and $ \Phi_3$ be Young functions and let $L^{\Phi_1}(\mathbb{R})$, $L^{\Phi_2}(\mathbb{R})$ and $L^{\Phi_3}(\mathbb{R})$ be the corresponding Orlicz spaces. We say that a function $m(\xi,\eta)$ defined on $\mathbb{R}\times \mathbb{R}$ is a bilinear multiplier of type $(\Phi_1,\Phi_2,\Phi_3)$ if \[ B_m(f,g)(x)=\int_\mathbb{R} \int_\mathbb{R} \hat{f}(\xi) \hat{g}(\eta)m(\xi,\eta)e^{2\pi i (\xi+\eta) x}d\xi d\eta \] defines a bounded bilinear operator from $L^{\Phi_1}(\mathbb{R}) \times L^{\Phi_2}(\mathbb{R})$ to $L^{\Phi_3}(\mathbb{R})$. We denote by $BM_{(\Phi_1,\Phi_2,\Phi_3)}(\mathbb{R})$ the space of all bilinear multipliers of type $(\Phi_1,\Phi_2,\Phi_3)$ and investigate some properties of such a class. Under some conditions on the triple $(\Phi_1,\Phi_2,\Phi_3)$ we give some examples of bilinear multipliers of type $(\Phi_1,\Phi_2,\Phi_3)$. We will focus on the case $m(\xi,\eta)=M(\xi-\eta) $ and get necessary conditions on $(\Phi_1,\Phi_2,\Phi_3)$ to get non-trivial multipliers in this class. In particular we recover some of the the known results for Lebesgue spaces.