Explicit upper bound for entropy numbers (Q1884375)

Let \(\Omega\) denote a bounded domain of \(\mathbb{R}^m\) with a Lipschitz boundary, let \(r\in\mathbb{N}\) and \(p\in(1,\infty)\), and let \(W^{r,p} (\Omega)\) be the associated Sobolev space normed by \[ \| u\|^p_{W^{r,p}(\Omega)}=\sum_{|\alpha|\leq r} \int|D^\alpha u |^p \] (where \(|\alpha|=\alpha_1+\cdots+\alpha_m\) is the `length' of the multi-index \(\alpha=(\alpha_1,\dots,\alpha_m)\in\mathbb{N}_0^m)\). Let \(I:W^{r,p}(\Omega)\to C(\overline\Omega)\) be the natural embedding of \(W^{r,p} (\Omega)\) into the space of continuous functions on \(\overline\Omega\). Let \(\varepsilon_k(IB_W)\) denote the \(k\)th entropy number of the image \(IB_W\) of the unit ball of \(W\). Let \(e_k= \varepsilon_{2^{k-1}}\) and \(e_k(I:W\to C)=e_k(IB_W)\). The paper is concerned with the cube \(\Omega=(-l,l)^m\) and the case \(rp>m\); it determines positive constants \(\alpha=\alpha(r,p,m,l)\) and \(\beta=\beta (r,p,m, l)\) such that, for \(k>\beta\), \[ e_k(K:W^{r,p}\bigl((-l,l)^m\bigr) \to C\bigl([-l, l]^m\bigr)\leq\alpha(k-\beta)^{-r/m}. \] The proof follows the ideas of \textit{M. S. Birman} and \textit{M. Z. Solomyak} [AMS (Translations, Series 2, 114, Providence, RI:AMS) (1980; Zbl 0426.46020)]. \textit{F. Cucker} and \textit{S. Smale} [Bull. Am. Math Soc., New Ser. 39, No. 1, 1--49 (2002); Zbl 0983.68162)] asked for explicit bounds for entropy numbers of embeddings in a more general context; this paper provides an answer for the Sobolev space \(W^{r,p}(\Omega)\) in the case that \(\Omega\) is a cube.