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Keywords:

  • Entropy numbers;
  • sequence spaces;
  • vector-valued sequence spaces;
  • Besov spaces;
  • interpolation;
  • msc (2010) 41A46;
  • 46B70;
  • 46E30;
  • 46E35;
  • 46E40;
  • 47B06

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 equation image in 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.