On Function Spaces with Wavelet Transform in Lpω(ℝd × ℝ+)

Abstract

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