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

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 $0<p<rleq infty$ and $0<qleq uleq infty$. 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.