# Estimates of entropy numbers of operators acting between function spaces

David E. Edmunds, Y Netrusov

Research output: Contribution to journalArticle (Academic Journal)peer-review

## 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 $B_{p_{1},\theta _{1}}^{\omega _{1}}(Q)$equation image in $B_{p_{2},\theta _{2} }^{\omega _{2}}(Q),$equation image where Q = (0, 1)d; θ1, θ2, p1, p2 ∈ (0, ∞], when the condition 1/θ1 − 1/θ2 ≥ 1/p1 − 1/p2 > 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.
Translated title of the contribution Estimates of entropy numbers of operators acting between function spaces English 614-630 17 Mathematische Nachrichten 286 5-6 https://doi.org/10.1002/mana.201100195 Published - 2 Apr 2013