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

We study the embedding id:lbp(ldq)RTlbr(ldu) and prove matching bounds for the entropy numbers ek(id) provided that 0<p<r<IF and 0<q<u<IF. 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 Dung, Temlyakov, and Ullrich. Both results rely on a novel covering construction recently found by Edmunds and Netrusov.