A note on embedding inequalities for weighted Sobolev and Besov spaces (Q2119705)

Let \(H^d\), \(0<d<n\), be the Hausdorff capacity in \(\mathbb R^n\). The limiting embeddings \[ \int_{\mathbb R^n} |f| \, d H^{n-k} \le c \, \|f \, | \dot{W}^k_1 (\mathbb R^n) \|, \qquad f \in \mathscr D(\mathbb R^n), \] \(1\le k <n\), \(k\in \mathbb N\), for the related homogeneous Sobolev spaces and its generalization \[ \int_{\mathbb R^n} |f| \, d H^{n-s} \le c \,\|f \, | \dot{B}^s_{1,1} (\mathbb R^n) \|, \qquad f\in \mathscr D(\mathbb R^n), \] \(0<s<n\), for the related homogeneous Besov spaces go back to \textit{D. R. Adams} [Lect. Notes Math. 1302, 115--124 (1988; Zbl 0658.31009)] and \textit{J. Xiao} [Adv. Math. 207, No. 2, 828--846 (2006; Zbl 1104.46022)]. The paper deals with weighted generalizations of these assertions both for weighted Hausdorff capacities and weighted Sobolev-Besov spaces where the weights belong to the Muckenhoupt class~\(A_1\).