Abstract
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 \(w(x) = \exp \left( {\sqrt {\log (1 + |x|)} } \right)\). For such weights we prove almost optimal estimates of the entropy numbers e k (id: \(B_{p_2 q_1 }^{s_1 } (\mathbb{R}^d ,w) \to B_{p_2 q_2 }^{s_2 } (\mathbb{R}^d )\)).
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References
B. Carl and I. Stephani, Entropy, Compactness and the Approximation of Operators (Cambridge Univ. Press, Cambridge, 1990).
D. E. Edmunds and H. Triebel, Function Spaces, Entropy Numbers, Differential Operators (Cambridge Univ. Press, Cambridge, 1996).
D. Haroske, “Embeddings of Some Weighted Function Spaces on ℝn; Entropy and Approximation Numbers. A Survey of Some Recent Results,” An. Univ. Craiova, Ser. Mat. Inform. 24, 1–44 (1997).
D. D. Haroske and H. Triebel, “Wavelet Bases and Entropy Numbers in Weighted Function Spaces,” Math. Nachr. 278, 108–132 (2005).
H. König, Eigenvalue Distribution of Compact Operators (Birkhäuser, Basel, 1986).
T. Kühn, “A Lower Estimate for Entropy Numbers,” J. Approx. Theory 110, 120–124 (2001).
T. Kühn, “Entropy Numbers of General Diagonal Operators,” Rev. Mat. Complut. 18, 479–491 (2005).
T. Kühn, H.-G. Leopold, W. Sickel, and L. Skrzypczak, “Entropy Numbers of Embeddings of Weighted Besov Spaces,” Constr. Approx. 23, 61–77 (2006).
T. Kühn, H.-G. Leopold, W. Sickel, and L. Skrzypczak, “Entropy Numbers of Embeddings of Weighted Besov Spaces. II,” Proc. Edinb. Math. Soc. 49, 331–359 (2006).
T. Kühn, H.-G. Leopold, W. Sickel, and L. Skrzypczak, “Entropy Numbers of Embeddings of Weighted Besov Spaces. III: Weights of Logarithmic Type,” Math. Z. 255, 1–15 (2007).
J. Peetre, New Thoughts on Besov Spaces (Duke Univ. Press, Durham, 1976).
A. Pietsch, Operator Ideals (VEB Deutscher Verlag der Wissenschaften, Berlin, 1978; North-Holland, Amsterdam, 1980).
C. Schütt, “Entropy Numbers of Diagonal Operators between Symmetric Banach Spaces,” J. Approx. Theory 40, 121–128 (1984).
H. Triebel, Theory of Function Spaces (Birkhäuser, Basel, 1983).
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Dedicated to Sergeĭ Mikhaĭlovich Nikol’skiĭ on the occasion of his 100th birthday
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Kühn, T. Entropy numbers in weighted function spaces. The case of intermediate weights. Proc. Steklov Inst. Math. 255, 159–168 (2006). https://doi.org/10.1134/S0081543806040134
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DOI: https://doi.org/10.1134/S0081543806040134