Entropy numbers of embeddings of Sobolev spaces in Zygmund spaces

D. Edmunds; Yu. Netrusov

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Let $E$ be a self-similar set with similarities ratio $r_{j} (0 \leq j \leq m - 1)$ and Hausdorff dimension $s$ , let $\vec{p} (p_{0}, p_{1}) ... p_{m - 1}$ be a probability vector. The Besicovitch-type subset of $E$ is defined as $E (\vec{p}) = \{x \in E : lim_{n \to \infty} \frac{1}{n} \sum_{k = 1}^{n} χ_{j} (x_{k}) = p_{j}, 0 \leq j \leq m - 1\},$ where $χ_{j}$ is the indicator function of the set ${j}$ . Let $α = {dim}_{H} (E (\vec{p})) = {dim}_{P} (E (\vec{p})) = \frac{\sum_{j = 0}^{m - 1} p_{j} log p_{j}}{\sum_{j = 0}^{m - 1} p_{i} log r_{j}}$ and $g$ be a gauge function, then we prove in this paper:(i) If $\vec{p} = (r_{0}^{s}, r_{1}^{s}, \dots, r_{m - 1}^{s})$ , then $ℋ^{s} (E (\vec{p})) = ℋ^{s} (E), 𝒫^{s} (E (\vec{p})) = 𝒫^{s} (E),$ moreover both of $ℋ^{s} (E)$ and $𝒫^{s} (E)$ are finite positive;(ii) If $\vec{p}$ is a positive probability vector other than $(r_{0}^{s}, r_{1}^{s}, \dots, r_{m - 1}^{s})$ , then the gauge functions can be partitioned as follows ...

On generalized exponential functions

J. Mikusiński (1953)

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