Haroske, Dorothee D.; Skrzypczak, Leszek Entropy numbers of embeddings of function spaces with Muckenhoupt weights. III: Some limiting cases. (English) Zbl 1253.46043 J. Funct. Spaces Appl. 9, No. 2, 129-178 (2011). Let \(\{\varphi_j \}^\infty_{j=0}\) be the usual dyadic resolution of the unity in \(\mathbb{R}^n\). Let \(w \in {\mathcal A}_\infty\) be a Muckenhoupt weight. Then \(B^s_{p,q} (\mathbb{R}^n,w)\) with \(0<p,q \leq \infty\), \(s\in \mathbb{R}\), collects all \(f\in S' (\mathbb{R}^n)\) such that \[ \|f \, | B^s_{p,q} (\mathbb{R}^n,w) \| = \Big( \sum^\infty_{j=0} 2^{jsq} \big\| (\varphi_j \widehat{f} )^\vee \, | L_p (\mathbb{R}^n,w) \|^q \Big)^{1/q} \] is finite. There is an \(F\)-counterpart. The paper concentrates on the special case \[ w = w_{\alpha, \beta} (x) =|x|^{\alpha_1} \big( 1 - \log |x|\big)^{\alpha_2} \text{if \(|x| \leq 1\)}, \]\[ w= w_{\alpha,\beta}(x) = |x|^{\beta_1} \big( 1 + \log |x| \big)^{\beta_2} \text{if \(|x| > 1\)}, \] \(\alpha_1 > -n\), \(\beta_1 >-n\), \(\alpha_2 \in \mathbb{R}\), \(\beta_2 \in \mathbb{R}\). The first main result characterizes under which conditions the embedding \[ B^{s_1}_{p_1,q_1} \big( \mathbb{R}^n, w_{\alpha, \beta} \big) \hookrightarrow B^{s_2}_{p_2, q_2} (\mathbb{R}^n) \] is continuous and under which conditions this embedding is compact. The second main result characterizes the compactness in terms of entropy numbers. This is based on wavelet characterizations of these spaces and the reduction of these embeddings to corresponding embeddings between related sequence spaces.For parts I and II, see, respectively, [Ann. Acad. Sci. Fenn., Math. 36, No. 1, 111–138 (2011; Zbl 1222.46027)] and [Rev. Mat. Complut. 21, No. 1, 135–177 (2008; Zbl 1202.46039)]. Reviewer: Hans Triebel (Jena) Cited in 27 Documents MSC: 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 47B06 Riesz operators; eigenvalue distributions; approximation numbers, \(s\)-numbers, Kolmogorov numbers, entropy numbers, etc. of operators Keywords:Besov spaces; Triebel-Lizorkin spaces; Muckenhoupt weights; compact embeddings; entropy numbers Citations:Zbl 1222.46027; Zbl 1202.46039 PDFBibTeX XMLCite \textit{D. D. Haroske} and \textit{L. Skrzypczak}, J. Funct. Spaces Appl. 9, No. 2, 129--178 (2011; Zbl 1253.46043) Full Text: DOI