ON FUNCTION SPACES WITH WAVELET TRANSFORM IN L-omega(p) (R-d x R+)

Abstract

Let omega(1) and omega(2) be weight functions on R-d, R-d x R+, respectively. Throughout this paper, we define D-omega 1,omega 2(p,q) (R-d) to be the vector space of f is an element of L-omega 1(p) (R-d) such that the wavelet transform W-g(f) belongs to L-omega 2(q) (R-d x R+) for 1 <= p,q < infinity, where 0 not equal g is an element of S (R-d). We endow this space with a sum norm and show that D-omega 1,omega 2(p,q) (R-d) becomes a Banach space. We discuss inclusion properties, and compact embeddings between these spaces and the dual of D-omega 1,omega 2(p,q) (R-d). Later we accept that the variable s in the space D-omega 1,omega 2(p,q) (R-d) is fixed. We denote this space by (D-omega 1,omega 2(p,q) )(s) (R-d), and show that under suitable conditions (D-omega 1,omega 2(p,q))(s) (R-d) is an essential Banach Module over L-omega 1(1) (R-d). We obtain its approximate identities. At the end of this work we discuss the multipliers from (D-omega 1,omega 2(p,q))(s) (R-d) into L-omega 1-1(infinity) (R-d), and from L-omega 1(1) (R-d) into (D-omega 1,omega 2(p,q))(s) (R-d).