Geodesic
Entropy Numbers in Weighted Function Spaces. The Case of Intermediate Weights
Informatics and Automation, Function spaces, approximation theory, and nonlinear analysis, Tome 255 (2006), pp. 170-179

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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 $e_k\bigl (\mathrm{id}:B^{s_1}_{p_1 q_1}(\mathbb R^d,w)\to B^{s_2}_{p_2 q_2}(\mathbb R^d)\bigr)$. 
@article{TRSPY_2006_255_a12,
     author = {T. K\"uhn},
     title = {Entropy {Numbers} in {Weighted} {Function} {Spaces.} {The} {Case} of {Intermediate} {Weights}},
     journal = {Informatics and Automation},
     pages = {170--179},
     publisher = {mathdoc},
     volume = {255},
     year = {2006},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2006_255_a12/}
}
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T. Kühn. Entropy Numbers in Weighted Function Spaces. The Case of Intermediate Weights. Informatics and Automation, Function spaces, approximation theory, and nonlinear analysis, Tome 255 (2006), pp. 170-179. http://geodesic.mathdoc.fr/item/TRSPY_2006_255_a12/