Entropy Numbers in Weighted Function Spaces. The Case of Intermediate Weights

The exact asymptotic behavior of the entropy numbers of compact embeddings of weighted Besov spaces is known in many cases, in particular for power-type weights and logarithmic weights. Here we consider intermediate weights that are strictly between these two scales; a typical example is $w(x)=\exp\bigl(\sqrt {\log (1+|x|)}\,\bigr)$. For such weights we prove almost optimal estimates of the entropy numbers $e_k\bigl (\mathrm{id}:B^{s_1}_{p_1 q_1}(\mathbb R^d,w)\to B^{s_2}_{p_2 q_2}(\mathbb R^d)\bigr)$.