Revista Matemática Iberoamericana

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Volume 22, Issue 1, 2006, pp. 305–322
DOI: 10.4171/RMI/457

Published online: 2006-04-30

Compact embeddings of Brézis-Wainger type

Fernando Cobos[1], Thomas Kühn[2] and Tomas Schonbek[3]

(1) Universidad Complutense de Madrid, Spain
(2) Universität Leipzig, Germany
(3) Florida Atlantic University, Boca Raton, USA

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.

Keywords: Entropy numbers, compact embeddings, Besov spaces, Lipschitz spaces

Cobos Fernando, Kühn Thomas, Schonbek Tomas: Compact embeddings of Brézis-Wainger type. Rev. Mat. Iberoam. 22 (2006), 305-322. doi: 10.4171/RMI/457