Notes on bilinear multipliers on Orlicz spaces

2019/02/04 by Blasco, Oscar, Osancliol, Alen
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1902.01116

Abstract

Let Φ₁ , Φ₂ and Φ₃ be Young functions and let L^Φ₁(ℝ), L^Φ₂(ℝ) and L^Φ₃(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 B_m(f,g)(x)=∫_ℝ ∫_ℝ f(ξ) g(η)m(ξ,η)e^{2πi (ξ+η) x}dξdη defines a bounded bilinear operator from L^Φ₁(ℝ) × L^Φ₂(ℝ) to L^Φ₃(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 the known results for Lebesgue spaces.

Related

• Notes on the spaces of bilinear multipliers
• Bilinear multipliers on Lorenzt spaces
• Multipliers for Hardy-Orlicz spaces and applications
• Unimodular bilinear Fourier multipliers on L^p spaces
• Pointwise multipliers of Orlicz function spaces and factorization