Entropy numbers of operators acting between vector‐valued sequence spaces

Abstract

Abstract Entropy numbers of operators acting between vector‐valued sequence spaces are estimated using information about the coordinate mappings. To do this some new ideas of combinatorial type are used. The results are applied to give sharp two‐sided estimates of the entropy numbers of some embeddings of Besov spaces. For instance, our main result allows us to give exact two‐sided estimates of the entropy numbers of the natural embedding of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$B_{p_{1},\theta _{1}}^{\omega _{1}}(Q)$\end{document} in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$B_{p_{2},\theta _{2} }^{\omega _{2}}(Q),$\end{document} where Q = (0, 1) d ; θ 1 , θ 2 , p 1 , p 2 ∈ (0, ∞], when the condition 1/θ 1 − 1/θ 2 ≥ 1/ p 1 − 1/ p 2 > 0 is satisfied. This work enables us to construct an example showing that the behaviour under real interpolation of entropy numbers can be even worse than in the example of 7 .