Entropy numbers of embeddings of weighted Besov spaces (Q816973)

Let be given \(d\in N\), \(\alpha\in R\), \(0<p_1,p_2,q_1,q_2\leq \infty\), \(-\infty< s_2<s_1<\infty\). For each \(n\in N\), \(e_n\) be the \(n\)th entropy number of the embedding \(B^{s_1}_{p_1,q_1} (R^d, \alpha)\hookrightarrow B^{s_2}_{p_2,q_2}(R^d)\). Here \(B^{s_1}_{p_1,q_1}(R^d,\alpha)\) denotes the weighted Besov space whose weight is given by \(w_\alpha(x):=(1+|x|^2)^{\alpha/2}\) and \(B^{s_2}_{p_2,q_2} (R^d)\) denotes the unweighted Besov space, respectively. Under the assumption that \[ \alpha=\left(s_1-\frac{d} {p_1}\right)-\left(s_2-\frac{d}{p_2}\right)>d \max\left(0,\frac{1}{p_2} -\frac{1}{p_1}\right), \] it is shown that the number \[ \tau:=\frac{s_1-s_2}{d}+\frac{1}{q_2}-\frac{1}{q_1} \] has the following properties: (i) if \(\tau>0\), then \(e_n\sim n^{-(s_1-s_2)/d}(1+\log n)^\tau\); (ii) if \(\tau=0\), then \(n^{-(s_1-s_2)/d}\preceq e_n\preceq n^{-(s_1-s_2)/d}(1+\log\log n)^{1/q_1}\); (iii) if \(\tau<0\), then \(e_n\sim n^{-(s_1-s_2)/d}\).