Let id be the natural embedding of the Sobolev space in the Zygmund space , where , 1 < p < ∞, l ∈ ℕ, 1/p = 1/q + l/n and a < 0, a ≠ -l/n. We consider the entropy numbers of this embedding and show that , 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.
@article{bwmeta1.element.bwnjournal-article-smv128i1p71bwm, author = {D. Edmunds and Yu. Netrusov}, title = {Entropy numbers of embeddings of Sobolev spaces in Zygmund spaces}, journal = {Studia Mathematica}, volume = {129}, year = {1998}, pages = {71-102}, zbl = {0919.46024}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv128i1p71bwm} }
Edmunds, D.; Netrusov, Yu. Entropy numbers of embeddings of Sobolev spaces in Zygmund spaces. Studia Mathematica, Tome 129 (1998) pp. 71-102. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv128i1p71bwm/
[00000] [BeS] C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, New York, 1988. | Zbl 0647.46057
[00001] [BiS1] M. S. Birman and M. Z. Solomyak, Piecewise polynomial approximations of functions of the class , Mat. Sb. 73 (1967), 331-355 (in Russian); English transl.: Math. USSR-Sb. (1967), 295-317. | Zbl 0173.16001
[00002] [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. | Zbl 0749.35026
[00003] [ET] D. E. Edmunds and H. Triebel, Function Spaces, Entropy Numbers, Differential Operators, Cambridge Univ. Press, Cambridge, 1996. | Zbl 0865.46020
[00004] [FJ] M. Frazier and B. Jawerth, A discrete transform and decompositions of distribution spaces, J. Funct. Anal. 93 (1990), 34-170. | Zbl 0716.46031
[00005] [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.
[00006] [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.
[00007] [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. | Zbl 0627.46038
[00008] [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.
[00009] [T] H. Triebel, Theory of Function Spaces II, Birkhäuser, Basel, 1992. | Zbl 0763.46025
[00010] [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.