Abstract
We study the embedding mathrm {id}: ell _p^b(ell _q^d) rightarrow ell _r^b(ell _u^d) and prove matching bounds for the entropy numbers e_k(mathrm {id}) provided that 0<p<rle infty and 0<qle ule infty . 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.
Highlights
We study the embedding id : b p and prove matching bounds for the entropy numbers ek(id) provided that 0 < p < r ≤ ∞ and 0 < q ≤ u ≤ ∞
Entropy numbers quantify the degree of compactness of a set, i.e., how well the set can be approximated by a finite set
Given a compact set K in a quasi-Banach space Y, the k-th entropy number ek(K, Y ) is defined to be the smallest radius ε > 0 such that K can be covered with 2k−1 copies of the ball ε BY, i.e., 2k−1 ek (K, Y ) := inf ε > 0 : ∃y1, . . . , y2k−1 such that K ⊂ y + ε BY, k ∈ N
Summary
Entropy numbers quantify the degree of compactness of a set, i.e., how well the set can be approximated by a finite set. Given a compact operator T : X → Y , where X and Y are quasi-Banach spaces, the k-th entropy number of the operator T is defined to be ek(T : X → Y ) := ek(T (BX ), Y ). One observes asymptotic decays of the form em (Id) n m−(r0−r1)(log m)(n−1)η, where η > 0 This behavior is well-known for s-numbers of these embeddings such as approximation, Gelfand, or Kolmogorov numbers, see [8] and the references therein. The crucial work has already been done in a recent work by Edmunds and Netrusov [10] They prove a general abstract version of Schütt’s theorem for operators between vector-valued sequence spaces.
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