Compact embeddings of Brézis-Wainger type

Abstract

Let $\Omega$ be a bounded domain in ${\mathbb{R}}^n$ and denote by $i{d}_{\Omega }$ the restriction operator from the Besov space $B_{pq}^{1+n/p}({\mathbb{R}}^n)$ into the generalized Lipschitz space $Li{p}^{(1,-\alpha )}(\Omega )$. We study the sequence of entropy numbers of this operator and prove that, up to logarithmic factors, it behaves asymptotically like $e_k(i{d}_{\Omega })\sim k^{-1/p}$ if $\alpha >\max (1+2/p-1/q,1/p)$. Our estimates improve previous results by Edmunds and Haroske.