Entropy numbers of embeddings of weighted Besov spaces III.: Weights of logarithmic type

We determine the exact asymptotic order of the entropy numbers of compact embeddings B-p1,q1(s1) (R-d, w(1)) hooked right arrow B-p2,q2(s2) (R-d, w(2)) of weighted Besov spaces in the case where the ratio of the weights w(x) = Wl(X)IW2(X) is Of logarithmic type. This complements the known results for weights of polynomial type. The estimates are given in terms of the number 1/p = 1/p(1) - 1/p(2) and the function w(x). We find an interesting new effect: if the growth rate at infinity of w(x) is below a certain critical bound, then the entropy numbers depend only on w(x) and no longer on the parameters of the two Besov spaces. All results remain valid for Triebel-Lizorkin spaces as well.