Compact embeddings of Brézis-Wainger type

Abstract

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