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 \[ B_m(f,g)(x)=\int_\mathbb{R} \int_\mathbb{R} \hat{f}(ξ) \hat{g}(η)m(ξ,η)e^{2πi (ξ+η) x}dξdη\] 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 $B{M}_{({\Phi }_1,{\Phi }_2,{\Phi }_3)}(\mathbb{R})$ 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.