Bilinear multipliers of small Lebesgue spaces

Abstract

Let G be a compact abelian metric group with Haar measure λ andĜ its dual with Haar measure µ .2, 3) be small Lebesgue spaces. A bounded sequence m (ξ, η) defined onĜ ×Ĝ 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∈Ĝ t∈Ĝf (s)ĝ (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 BM θ [(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 BM θ [(p 1 ; (p 2 ; (p 3 ] and give examples of bilinear multipliers.