On function spaces with wavelet transform in Lp?(?d × ?+)

Abstract

Let ?1 and ?2 be weight functions on ?d, ?d × ?+, respectively. Throughout this paper, we define (?d) to be the vector space of f ? (?d) such that the wavelet transform Wgf belongs to (?d × ?+) for 1 ? p, q < ?, where 0 6= g ? S (?d). We endow this space with a sum norm and show that (?d) becomes a Banach space. We discuss inclusion properties, and compact embeddings between these spaces and the dual of (?d). Later we accept that the variable s in the space (?d) is fixed. We denote this space by (?d), and show that under suitable conditions (?d) is an essential Banach Module over (?d). We obtain its approximate identities. At the end of this work we discuss the multipliers from (?d) into (?d), and from (?d) into (?d).