### Abstract:

Let ω1 and ω2 be weight functions on R d , R d × R+, respectively. Throughout this paper, we define D p,q ω1,ω2 R d to be the vector space of f ∈ L p ω1 R d such that the wavelet transform Wgf belongs to L q ω2 R d × R+ for 1 ≤ p,q < ∞, where 0 6= g ∈ S R d. We endow this space with a sum norm and show that Dp,q ω1,ω2 R d becomes a Banach space. We discuss inclusion properties, and compact embeddings between these spaces and the dual of D p,q ω1,ω2 R d . Later we accept that the variable s in the space D p,q
ω1,ω2 R d is fixed. We denote this space by D p,q ω1,ω2 s R d, and show that under suitable conditions Dp,q ω1,ω2 s R d is an essential Banach Module over L1 ω1 R
d. We obtain its approximate identities. At the end of this work we discuss the multipliers from D p,q ω1,ω2 s R d into L∞ω−11Rd , and from L 1 ω1 R d into D p,q ω1,ω2 s R d
.