Bilinear Multipliers of Weighted Lebesgue Spaces and Variable Exponent Lebesgue Spaces

Abstract

Let 1 ≤ p1, p2 <8, 0 < p3≤8and ω1, ω2, ω3 be weight functions on Rn. Assume that ω1, ω2 are slowly increasing functions. We say that a bounded function m(ξ,ν) defined on Rn × Rn is a bilinear multiplier on Rn of type (p1,ω1; p2,ω2; p3,ω3) (shortly (ω1,ω2,ω3)) if Bf,g)(x) = ∫∫f̂ (ξ)ĝ(η)m(ξ,η)e 2πi<ξ+ν,x> dξ dν is a bounded bilinear operator from Lp1,(Rn)×Lp2 ?2 (Rn) to Lp3 ?3 (Rn). We denote by BM(p1,ω1; p2,ω2; p3,ω3) (shortly BM(,ω1,ω2, ω3)) the vector space of bilinear multipliers of type (,ω1,ω2, ω3). In this paper first we discuss some properties of the space BM(,ω1,ω2,ω3). Furthermore, we give some examples of bilinear multipliers. At the end of this paper, by using variable exponent Lebesgue spaces Lp1(x)(Rn), Lp2(x)(Rn) and Lp3(x)(Rn), we define the space of bilinear multipliers from Lp1(x)(Rn)×Lp2(x)(Rn) to Lp3(x)(Rn) and discuss some properties of this space. © 2013 Kulak and Gürkanli; licensee Springer.