Compact embeddings of the spaces \(A^p_{w,\omega}(\mathbb R^d)\)
In the paper under review the author discusses the compact embeddings of some function spaces on \(\mathbb{R}^d\) into other function spaces on \(\mathbb{R}^d\). We describe some of these results. Let \(w\) be a Beurling weight function on \(\mathbb{R}^d\). For \(1 \leq p \leq \infty\) set \[ L_w^p (\mathbb{R}^d) = \{ f : fw \in L^p( \mathbb{R}^d) \}. \] For \(f \in L_w^1(\mathbb{R}^d), \hat{f}\) will denote the Fourier transform of \(f\). Let \(w, \omega\) be Beurling weights on \(\mathbb{R}^d\) and let \(A_{w, \omega}^p( \mathbb{R}^d)\) be the set that consists of all functions \(f \in L_w^1(\mathbb{R}^d)\) for which \(\hat{f} \in L_{\omega}^p (\mathbb{R}^d)\). The set \(A_{w, \omega}^p (\mathbb{R}^d)\) is a Banach space under a certain norm. One theorem proved in this paper is: Let \(w, \omega, v\) be Beurling weight functions on \(\mathbb{R}^d\). If \(v \preceq w\) and \(\frac{v(x)}{w(x)}\) does not tend to zero in \(\mathbb{R}^d\) as \(x \rightarrow \infty\), then the embedding of the space \(A_{w, \omega}^p(\mathbb{R}^d)\) into \(L_v^1 (\mathbb{R}^d)\) is never compact. For \(1 \leq p \leq \infty\) let \(\Lambda_{w, \omega}^p (\mathbb{R}^d)\) denote the set of all functions in \(L_w^1 (\mathbb{R}^d)\) for which \(\hat{f} \in L_{\omega}^1 (\mathbb{R}^d) \cap L_{\omega}^p (\mathbb{R}^d).\) Now \(\Lambda_{w, \omega}^p (\mathbb{R}^d)\) is a Banach space under a certain norm. Another theorem proved in this paper by the author is: Let \(w_1, w_2\) and \(\omega_1, \omega_2\) be Beurling weight functions on \(\mathbb{R}^d\). Assume that \(\omega_1, \omega_2\) are symmetric on \(\mathbb{R}^d\) and \(w_2 \preceq w_1, \omega_2 \preceq \omega_1\). Then the embedding \[ i : \Lambda_{w_1, \omega_1}^p (\mathbb{R}^d) \mapsto \Lambda_{w_2, \omega_2}^p (\mathbb{R}^d) \] is compact if and only if \(\frac{w_2(x)}{w_1(x)}\) and \(\frac{\omega_2(x)}{\omega_1(x)}\) tend to zero.
scientific article; zbMATH DE number 5548633