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 $f in L^p_{omega_1} (Bbb{R}^d)$ such that the wavelet transform $W_gf$ belongs to $L^q_{omega_2} (Bbb{R}^d X Bbb{R}_+)$ 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 $L^1_{omega_1} (Bbb{R}^d)$ . 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 $L^1_{omega_1} (Bbb{R}^d)$ into $(D^{p,q}_{omega_1,omega_2})_s (Bbb{R}^d)$