COMPACT EMBEDDINGS OF THE SPACES Ap w,ω

Abstract

For $1\leq p\leq \infty , A_{w,\omega }^{p}\left( R^{d}\right) $ denotes the space (Banach space) of all functions in $L_{w}^{1}\left( R^{d}\right) $ a weighted $L^{1}-$space (Beurling algebra) with Fourier transforms $\overset{\wedge }{f}$ in\ $L_{\omega }^{p}\left( R^{d}\right) $ which is equipped with the sum norm \begin{equation*} \left\Vert f\right\Vert _{w,\omega }^{p}=\left\Vert f\right\Vert _{1,w}+\left\Vert \overset{\wedge }{f}\right\Vert _{p,\omega }, \end{equation*}% where $w$ and $\omega $ are Beurling weights on $R^{d}$.This space was defined in $\left[ 5\right] $ and generalized in $\left[ 6\right] .$ The present paper is a sequal to these works.In this paper we are going to discuss compact embeddings between the spaces $A_{w,\omega }^{p}\left( R^{d}\right) .$