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

Abstract

We determine the exact asymptotic order of the entropy numbers of compact embeddings \(B^{s_1}_{p_1 , q_1}(\mathbb{R}^{d},w_1)\hookrightarrow B^{s_2}_{p_2 , q_2}(\mathbb{R}^{d},w_2)\) of weighted Besov spaces in the case where the ratio of the weights w(x) = w 1(x)/w 2(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.