Entropy numbers of embeddings of Sobolev spaces in Zygmund spaces

D. Edmunds; Yu. Netrusov

Studia Mathematica (1998)

  • Volume: 128, Issue: 1, page 71-102
  • ISSN: 0039-3223

Abstract

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Let id be the natural embedding of the Sobolev space W p l ( Ω ) in the Zygmund space L q ( l o g L ) a ( Ω ) , where Ω = ( 0 , 1 ) n , 1 < p < ∞, l ∈ ℕ, 1/p = 1/q + l/n and a < 0, a ≠ -l/n. We consider the entropy numbers e k ( i d ) of this embedding and show that e k ( i d ) ≍ k - η , where η = min(-a,l/n). Extensions to more general spaces are given. The results are applied to give information about the behaviour of the eigenvalues of certain operators of elliptic type.

How to cite

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Edmunds, D., and Netrusov, Yu.. "Entropy numbers of embeddings of Sobolev spaces in Zygmund spaces." Studia Mathematica 128.1 (1998): 71-102. <http://eudml.org/doc/216476>.

@article{Edmunds1998,
abstract = {Let id be the natural embedding of the Sobolev space $W_p^l(Ω)$ in the Zygmund space $L_q(log L)_a(Ω)$, where $Ω = (0,1)^n$, 1 < p < ∞, l ∈ ℕ, 1/p = 1/q + l/n and a < 0, a ≠ -l/n. We consider the entropy numbers $e_k(id)$ of this embedding and show that $e_k(id) ≍ k^\{-η\}$, where η = min(-a,l/n). Extensions to more general spaces are given. The results are applied to give information about the behaviour of the eigenvalues of certain operators of elliptic type.},
author = {Edmunds, D., Netrusov, Yu.},
journal = {Studia Mathematica},
keywords = {embedding; Sobolev space; Zygmund space; entropy numbers; eigenvalues; operators of elliptic type},
language = {eng},
number = {1},
pages = {71-102},
title = {Entropy numbers of embeddings of Sobolev spaces in Zygmund spaces},
url = {http://eudml.org/doc/216476},
volume = {128},
year = {1998},
}

TY - JOUR
AU - Edmunds, D.
AU - Netrusov, Yu.
TI - Entropy numbers of embeddings of Sobolev spaces in Zygmund spaces
JO - Studia Mathematica
PY - 1998
VL - 128
IS - 1
SP - 71
EP - 102
AB - Let id be the natural embedding of the Sobolev space $W_p^l(Ω)$ in the Zygmund space $L_q(log L)_a(Ω)$, where $Ω = (0,1)^n$, 1 < p < ∞, l ∈ ℕ, 1/p = 1/q + l/n and a < 0, a ≠ -l/n. We consider the entropy numbers $e_k(id)$ of this embedding and show that $e_k(id) ≍ k^{-η}$, where η = min(-a,l/n). Extensions to more general spaces are given. The results are applied to give information about the behaviour of the eigenvalues of certain operators of elliptic type.
LA - eng
KW - embedding; Sobolev space; Zygmund space; entropy numbers; eigenvalues; operators of elliptic type
UR - http://eudml.org/doc/216476
ER -

References

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  3. [BiS2] M. S. Birman and M. Z. Solomyak, Estimates for the number of negative eigenvalues of the Schrödinger operator and its generalizations, in: Adv. Soviet Math. 7, Amer. Math. Soc., 1991, 1-55. Zbl0749.35026
  4. [ET] D. E. Edmunds and H. Triebel, Function Spaces, Entropy Numbers, Differential Operators, Cambridge Univ. Press, Cambridge, 1996. Zbl0865.46020
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  8. [N1] Yu. V. Netrusov, Embedding theorems for Lizorkin-Triebel spaces, Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 159 (1987), 103-112 (in Russian); English transl.: Soviet Math. 47 (1989), 2896-2903. Zbl0627.46038
  9. [N2] Yu. V. Netrusov, The exceptional sets of functions from Besov and Lizorkin-Triebel spaces, Trudy Mat. Inst. Steklov. 187 (1989), 162-177 (in Russian); English transl.: Proc. Steklov Inst. Math. 187 (1990), 185-203. 
  10. [T] H. Triebel, Theory of Function Spaces II, Birkhäuser, Basel, 1992. Zbl0763.46025
  11. [V] A. G. Vitushkin, Estimation of the Complexity of the Tabulation Problem, Gos. Izdat. Fiz.-Mat. Lit., Moscow, 1959 (in Russian); English transl.: Theory of the Transmission and Processing of Information, Pergamon Press, New York, 1961. 

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