Entropy numbers of finite dimensional mixed-norm balls and function space embeddings with small mixed smoothness

Abstract

We study the embedding $\text{id}:{\ell }_p^b({\ell }_q^d)\to {\ell }_r^b({\ell }_u^d)$ and prove matching bounds for the entropy numbers $e_k(\text{id})$ provided that $0<p<r\le \infty$ and $0<q\le u\le \infty$. 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 settles an open question in the literature. Both results rely on a novel covering construction recently found by Edmunds and Netrusov.