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Embeddings and entropy numbers in Besov spaces of generalized smoothness. (English) Zbl 0966.46018
Hudzik, Henryk (ed.) et al., Function spaces. Proceedings of the 5th international conference, Poznań, Poland, August 28-September 3, 1998. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 213, 323-336 (2000).
The author is interested in Sobolev embeddings of functions spaces on bounded domains and their entropy numbers. It is well known that in the limiting case the classical embeddings are continuous but not compact. To improved the situation one can take a little larger target spaces or a little smaller initial space. The second case is investigated in the paper. The author regards the Besov spaces with generalized smoothness \(B^{s,b}_{p,q}\), \(s,b\in\mathbb{R}\) and \(0< p, q\leq\infty\). The norm in the last space if defined on \(\mathbb{R}^n\) is given by \[ \Biggl( \sum^\infty_{j=0} 2^{jsq}(1+ j)^{bq}\|(\varphi_i\widehat f)^\vee|L_p\|^q\Biggr)^{1/q}, \] with the usual meaning of the symbols in the above formula. On a bounded domain \(\Omega\) the spaces are defined by restriction.
The \(k\) entropy number \(e_k\) of the embedding is the infimum of all numbers \(\varepsilon> 0\) such that \(2^{k-1}\) ball in the arget space of radius \(\varepsilon\) cover the unit ball of the initial space. The convergence to zero of the sequence \(e_k\) is equivalent to the compactness of the embeddings. Different estimates of the entropy numbers \(e_k\) of embeddings of \(B^{s_1,b}_{p_1,q_1}(\Omega)\) into Besov spaces \(B^{s_2}_{p_2,q_2}(\Omega)\) are given. In particular it is proved that, the entropy number \(e_k\) can be estimated in the following way \[ ck^{-(s_1- s_2)/n}(\log k)^{-b}\leq e_k\leq Ck^{-(s_1- s_2)/n}(\log k)^{-b+ s^*}, \] if \(s_1< s_2\), \(p_1< p_2\), \(s_1- s_2= n(1/p_1- 1/p_2)\), \(b> (1/q_2- 1/q_1)_+\) and \(s^*= 2(s_1- s_2)/n+1/\text{min}(p_2\), \(q_2,1)\).
For the entire collection see [Zbl 0943.00053].

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