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 contributionEstimates of entropy numbers of operators acting between function spaces
Original languageEnglish
Pages (from-to)614-630
Number of pages17
JournalMathematische Nachrichten
Volume286
Issue number5-6
DOIs
Publication statusPublished - 2 Apr 2013

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