Entropy numbers of finite-dimensional Lorentz space embeddings

Abstract

The sequence of entropy numbers quantifies the degree of compactness of a linear operator acting between quasi-Banach spaces. We determine the asymptotic behavior of entropy numbers in the case of natural embeddings between finite-dimensional Lorentz spaces $\ell_{p,q}^n$ in all regimes; our results are sharp up to constants. This generalizes classical results obtained by Sch\"utt (in the case of Banach spaces) and Edmunds and Triebel, K\"uhn, as well as Gu\'edon and Litvak (in the case of quasi-Banach spaces) for entropy numbers of identities between finte-dimensional Lebesgue sequence spaces $\ell_p^n$. We employ techniques such as interpolation, volume comparison as well as techniques from sparse approximation and combinatorial arguments. Further, we characterize entropy numbers of embeddings between finite-dimensional symmetric quasi-Banach spaces in terms of best $s$-term approximation numbers.