Entropy numbers of limiting embeddings of logarithmic Sobolev spaces into exponential spaces
Let \(\Omega\) be a bounded domain in \(\mathbb R^n\), \(n \geq 2\), and let \(E_\nu (\Omega)\), \( \nu >0\), be the Orlicz space with Young function behaving like \(\exp (t^\nu)\) for large \(t\). Let \(W^s_p (\log W)_\alpha (\Omega)\) be the logarithmic Sobolev spaces, where the underlying space \(L_p (\Omega)\) is replaced by the Zygmund space \(L_p (\log L)_\alpha (\Omega)\). Let \(0 < \alpha < 1 - \frac{1}{n}\), \(\frac{1}{q_\alpha} = 1 - \frac{1}{n} - \alpha\) and \(0<\nu < q_\alpha.\) Then the embedding \[ W^1_n (\log W)_\alpha (\Omega) \hookrightarrow E_{q_\alpha} (\Omega) \] is continuous, but not compact, and the embeddings \[ W^1_n (\log W)_\alpha (\Omega) \hookrightarrow E_\nu (\Omega) \] are compact. The paper deals with estimates for the entropy numbers of these compact embeddings with Theorem 3.2 as the main result.
scientific article; zbMATH DE number 5689271
scientific article