Anisotropic Besov spaces and approximation numbers of traces on related fractal sets
Let \(B^{s,a}_{pq} (\mathbb R^n)\) be the well-known anisotropic Besov spaces in \(\mathbb R^n\), where \(a= (a_1, \dots, a_n)\) with \(0<a_1\leq\dots\leq a_n\), \(\sum^n_{l=1}a_l=1\), stands for the anisotropy and \(1<p<\infty\), \(s>0\). Let \(\mu\) be a compactly supported Radon measure in \(\mathbb R^n\) with \(\Gamma=\text{supp}\,\mu. \) It is the first aim of this paper to clarify under which circumstances the trace operator \[ \text{tr}_\Gamma:B^{s,a}_{pp} (\mathbb R^n)\hookrightarrow L_p (\Gamma) \] makes sense and is compact. If \(\Gamma\) is a compact anisotropic \(d\)-set then the author proves for some of these embeddings \[ a_k (\text{tr}_\Gamma)\sim k^{-\lambda}, \quad k \in\mathbb N, \] where \(a_k\) are the related approximation numbers and \(\lambda >0\). To get these results, some specific properties of anisotropic Besov spaces are derived, namely: homogeneity, adapted equivalent norms, and localisation.
scientific article; zbMATH DE number 5142568