ON FUNCTION SPACES WITH WAVELET TRANSFORM IN Lp ω(R^d × R+)

Let !1 and !2 be weight functions on R^d, R^d × R+, respectively. Throughout this paper, we define Dp,q ω1,ω2(R^d) to be the vector space of f ∈ Lp ω1(R^d) such that the wavelet transform Wgf belongs to Lq ω2(R^d × R+) for 1 ≤ p, q ∞, where 0 ≠ 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 emembeddings between these spaces and the dual of Dp,q ω1,ω2(R^d). Later we accept that the variable s in the space Dp,q ω1,ω2(R^d) is fixed. We denote this space by (Dpq ω1,ω2)s (R^d), and show that under suitable conditions (Dpq ω1,ω2) (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 (Dpq ω1,ω2)s (R^d) into L∞ ω−1 (R^d) , and from L∞ ω1 (R^d) into (Dpq ω1,ω2)s (R^d).