Bilinear multipliers of function spaces with wavelet transform in (Rn)

Let w1, w2 be weight functions on Rn. For 1 < p.q < 00, fixed s G R+, tlie space (Dpwq19,W2), (Rn) consists of / G (Rn) such that wavelet transform Wgf belongs to (Rn) where 0 / 3 G 67 (Rn). This space was defined and investigated by Kulak and Gurkanli 11. In this paper using this function space, the vector space of bilinear multipliers is defined in this way. Let w1, w2, v1, be slowly increasing weight functions and let oj3, z/3 be any weight functions on R77. Assume that m(£,rj) is a bounded, measurable function on R" x R". We define Bm(f,g) (x)= f ff(E)G(N)mforiedSdn for all f,g (Rn). We say that m (E, N) is a bilinear multiplier on R of type (D (pi,Qi,wi,Vi,SiVi,) if Bm is bounded operator from X (RN) to (Rn) where 1 < Pi, q1 < OO, Sit R+ (i = 1,2,3). We denote by BMD (pi, qi. cjj, si;) the vector space of bilinear multipliers of type (D (pi, qi,wi, vi, Si)). In this work, some properties of this space are investigated and some examples of these bilinear multipliers are given.