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

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• 作者：Thomas Kühn ; Hans-Gerd Leopold ; Winfried Sickel and Leszek Skrzypczak
• 关键词：Weighted Besov spaces ; Smooth weights with logarithmic bounds ; Continuous and compact embeddings ; Entropy numbers
• 刊名：Mathematische Zeitschrift
• 出版年：2007
• 出版时间：January, 2007
• 年：2007
• 卷：255
• 期：1
• 页码：1-15
• 全文大小：223 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-1823

文摘

We determine the exact asymptotic order of the entropy numbers of compact embeddings Bs_{1p1 , q1}(\mathbbRd,w₁)\hookrightarrow Bs_{2p2 , q2}(\mathbbRd,w₂)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.