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Entropy numbers of embeddings of function spaces with Muckenhoupt weights. III: Some limiting cases. (English) Zbl 1253.46043

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)].

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
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