On function spaces with wavelet transform in $L_{\omega}^p(\Bbb{R}^d X \Bbb{R}_+)$

Abstract

Let \omega_1 and \omega_2 be weight functions on \Bbb{R}d, (\Bbb{R}d X \Bbb{R}_), respectively. Throughout this paper, we define D{p,q}_{\omega_1,\omega_2} (\Bbb{R}d) to be the vector space of f \in Lp_{\omega_1} (\Bbb{R}d) such that the wavelet transform W_gf belongs to Lq_{\omega_2} (\Bbb{R}d X \Bbb{R}_) for 1 \leq p, q < \infty, where 0 \neq g \in S (\Bbb{R}d) . We endow this space with a sum norm and show that D{p,q}_{\omega_1,\omega_2} (\Bbb{R}d) becomes a Banach space. We discuss inclusion properties, and compact embeddings between these spaces and the dual of D{p,q}_{\omega_1,\omega_2} (\Bbb{R}d). Later we accept that the variable s in the space D{p,q}_{\omega_1,\omega_2} (\Bbb{R}d) is fixed. We denote this space by (D{p,q}_{\omega_1,\omega_2})_s (\Bbb{R}d) , and show that under suitable conditions (D{p,q}_{\omega_1,\omega_2})_s (\Bbb{R}d) is an essential Banach Module over L1_{\omega_1} (\Bbb{R}d) . We obtain its approximate identities. At the end of this work we discuss the multipliers from (D{p,q}_{\omega_1,\omega_2})_s (\Bbb{R}d) into L{\infty}_{(\omega_1){-1}} (\Bbb{R}d), and from L1_{\omega_1} (\Bbb{R}d) into (D{p,q}_{\omega_1,\omega_2})_s (\Bbb{R}d)