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

## TL;DR

This paper derives precise bounds for entropy numbers of finite-dimensional mixed-norm balls and applies these results to determine optimal asymptotic rates for embeddings of certain function spaces with small mixed smoothness, resolving an open problem.

## Contribution

It provides the first matching bounds for entropy numbers of mixed-norm embeddings and establishes optimal dimension-free rates for Besov and Triebel-Lizorkin space embeddings with small mixed smoothness.

## Key findings

- Matching bounds for entropy numbers of mixed-norm embeddings
- Optimal asymptotic rates for function space embeddings
- Resolution of an open problem in the literature

## 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 $0<p<r\leq \infty$ and $0<q\leq u\leq \infty$. 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.