Entropy numbers of embeddings of Sobolev spaces in Zygmund spaces

Let id be the natural embedding of the Sobolev space ${\displaystyle W_p^l(\Omega )}$ in the Zygmund space ${\displaystyle L_q{\left(\log L\right)}_a(\Omega )}$, where ${\displaystyle \Omega ={(0,1)}^n}$, 1 < p < ∞, l ∈ ℕ, 1/p = 1/q + l/n and a < 0, a ≠ -l/n. We consider the entropy numbers ${\displaystyle e_k(id)}$ of this embedding and show that ${\displaystyle e_k(id)\asymp k^{-\eta }}$, where η = min(-a,l/n). Extensions to more general spaces are given. The results are applied to give information about the behaviour of the eigenvalues of certain operators of elliptic type.