Notes on bilinear multipliers on Orlicz spaces

Let Φ1,Φ2 and Φ3 be Young functions and let LΦ1(R), LΦ2(R) and LΦ3(R) be the corresponding Orlicz spaces. We say that a function m(ξ,η) defined on R×R is a bilinear multiplier of type (Φ1,Φ2,Φ3) if Bm(f,g)(x)=∫R∫Rf̂(ξ)ĝ(η)m(ξ,η)e2πi(ξ+η)xdξdηdefines a bounded bilinear operator from LΦ1(R)×LΦ2(R) to LΦ3(R). We denote by BM(Φ1,Φ2,Φ3)(R) the space of all bilinear multipliers of type (Φ1,Φ2,Φ3) and investigate some properties of such a class. Under some conditions on the triple (Φ1,Φ2,Φ3) we give some examples of bilinear multipliers of type (Φ1,Φ2,Φ3). We will focus on the case m(ξ,η)=M(ξ−η) and get necessary conditions on (Φ1,Φ2,Φ3) to get non‐trivial multipliers in this class. In particular we recover some of the known results for Lebesgue spaces.