Bilinear multipliers of small Lebesgue spaces

Abstract

Let G be a compact abelian metric group with Haar measure λ and Gˆ its dual with Haar measure µ. Assume that 1 < pi < ∞, p ′ i = pi pi−1 , (i = 1, 2, 3) and θ ≥ 0. Let L (p ′ i ,θ (G), (i = 1, 2, 3) be small Lebesgue spaces. A bounded sequence m (ξ, η) defined on Gˆ × Gˆ is said to be a bilinear multiplier on G of type [(p ′ 1; (p ′ 2; (p ′ 3] θ if the bilinear operator Bm associated with the symbol m Bm (f, g) (x) = ∑ s∈Gˆ ∑ t∈Gˆ ˆf (s) ˆg (t) m (s, t)⟨s + t, x⟩ defines a bounded bilinear operator from L (p ′ 1 ,θ (G) × L (p ′ 2 ,θ (G) into L (p ′ 3 ,θ (G). We denote by BMθ [(p ′ 1; (p ′ 2; (p ′ 3] the space of all bilinear multipliers of type [(p ′ 1; (p ′ 2; (p ′ 3] θ . In this paper, we discuss some basic properties of the space BMθ [(p ′ 1; (p ′ 2; (p ′ 3] and give examples of bilinear multipliers.