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Abstrakty
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.
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
71-102
Opis fizyczny
Daty
wydano
1998
otrzymano
1997-07-07
Twórcy
autor
- School of Mathematical Sciences, University of Sussex, Falmer, Brighton BN1 9QHi, U.K., D.E.Edmunds@sussex.ac.uk
autor
- 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.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-smv128i1p71bwm