Entropy Numbers of Finite Dimensional Mixed-Norm Balls and Function Space Embeddings with Small Mixed Smoothness

Abstract

We study the embedding id:ℓpb(ℓqd)→ℓrb(ℓud) and prove matching bounds for the entropy numbers ek(id) provided that 0 < p< r≤ ∞ and 0 < q≤ u≤ ∞. Based on this finding, we establish optimal dimension-free asymptotic rates for the entropy numbers of embeddings of Besov and Triebel–Lizorkin spaces of small dominating mixed smoothness, which gives a complete answer to an open problem mentioned in the recent monograph by Dũng, Temlyakov, and Ullrich. Both results rely on a novel covering construction recently found by Edmunds and Netrusov.