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1998 | 128 | 1 | 71-102
Tytuł artykułu

Entropy numbers of embeddings of Sobolev spaces in Zygmund spaces

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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.
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  • King's College London, Strand, London WC2R 2LS, U.K.
Bibliografia
  • [BeS] C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, New York, 1988.
  • [BiS1] M. S. Birman and M. Z. Solomyak, Piecewise polynomial approximations of functions of the class $W_p^α$, Mat. Sb. 73 (1967), 331-355 (in Russian); English transl.: Math. USSR-Sb. (1967), 295-317.
  • [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.
  • [ET] D. E. Edmunds and H. Triebel, Function Spaces, Entropy Numbers, Differential Operators, Cambridge Univ. Press, Cambridge, 1996.
  • [FJ] M. Frazier and B. Jawerth, A discrete transform and decompositions of distribution spaces, J. Funct. Anal. 93 (1990), 34-170.
  • [KT] A. N. Kolmogorov and V. M. Tikhomirov, ε-entropy and ε-capacity of sets in functional spaces, Uspekhi Mat. Nauk 14 (2) (1959), 3-86 (in Russian); English transl.: Amer. Math. Soc. Transl. Ser. 2 17 (1961), 277-364.
  • [L] G. G. Lorentz, The 13th problem of Hilbert, in: Mathematical Developments Arising from Hilbert Problems, Proc. Sympos. Pure Math. 28, Amer. Math. Soc., 1976, 419-430.
  • [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.
  • [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.
  • [T] H. Triebel, Theory of Function Spaces II, Birkhäuser, Basel, 1992.
  • [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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