Abstract
For limiting noncompact Sobolev embeddings into continuous functions, we study the behavior of approximation, Gelfand, Kolmogorov, Bernstein and isomorphism s-numbers. In the one-dimensional case, the exact values of the above-mentioned strict s-numbers are obtained, and in the higher dimensions sharp estimates for asymptotic behavior of the strict s-numbers are established. As all known results for s-numbers of Sobolev type embeddings are studied mainly under the compactness assumption, our work is an extension of existing results and reveal an interesting behavior of s-numbers in the limiting case when some of them (approximation, Gelfand, and Kolmogorov) have positive lower bound and others (Bernstein and isomorphism) are decreasing to zero. It also follows from our results that such limiting noncompact Sobolev embeddings are finitely strictly singular maps.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alber, J., Niedermeier, R.: On multidimensional curves with Hilbert property. Theory Comput. Syst. 33(4), 295–312 (2000)
Bennett, C., Sharpley, R.: Interpolation of Operators. Pure and Applied Mathematics, vol. 129. Academic Press Inc., Boston (1988)
Bojarski, B.: Remarks on Sobolev imbedding inequalities. In Complex analysis, Joensuu 1987, vol. 1351 of Lecture Notes in Math., pp. 52–68. Springer, Berlin (1988)
Butz, A.R.: Convergence with Hilbert’s space filling curve. J. Comput. Syst. Sci. 3, 128–146 (1969)
Carl, B., Stephani, I.: Entropy, Compactness and the Approximation of Operators. Cambridge Tracts in Mathematics, vol. 98. Cambridge University Press, Cambridge (1990)
Chalendar, I., Fricain, E., Popov, A.I., Timotin, D., Troitsky, V.G.: Finitely strictly singular operators between James spaces. J. Funct. Anal. 256(4), 1258–1268 (2009)
Cianchi, A., Pick, L.: Sobolev embeddings into BMO, VMO, and \(L_\infty \). Ark. Mat. 36(2), 317–340 (1998)
Fabian, M., Habala, P., Hájek, P., Montesinos, V., Zizler, V.: Banach space theory. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. Springer, New York, The basis for linear and nonlinear analysis (2011)
Hencl, S.: Measures of non-compactness of classical embeddings of Sobolev spaces. Math. Nachr. 258, 28–43 (2003)
Lang, J., Edmunds, D.: Eigenvalues, embeddings and generalised trigonometric functions. Lecture Notes in Mathematics, vol. 2016. Springer, Heidelberg (2011)
Martio, O.: John domains, bi-Lipschitz balls and Poincaré inequality. Rev. Roum. Math. Pures Appl. 33(1–2), 107–112 (1988)
Moon, B., Jagadish, H.V., Faloutsos, C., Saltz, J.H.: Analysis of the clustering properties of the hilbert space-filling curve. IEEE Trans. Knowl. Data Eng. 13(1), 124–141 (2001)
Nguyen, V.K.: Bernstein numbers of embeddings of isotropic and dominating mixed Besov spaces. Math. Nachr. 288(14–15), 1694–1717 (2015)
Pietsch, A.: \(s\)-numbers of operators in Banach spaces. Studia Math. 51, 201–223 (1974)
Pietsch, A.: History of Banach Spaces and Linear Operators. Birkhäuser Boston Inc., Boston (2007)
Pinkus, A.: \(n\)-widths in approximation theory. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 7. Springer-Verlag, Berlin (1985)
Platzman, L.K., Bartholdi III, J.J.: Spacefilling curves and the planar travelling salesman problem. J. Assoc. Comput. Mach. 36(4), 719–737 (1989)
Reshetnyak, Y.G.: Integral representations of differentiable functions in domains with a nonsmooth boundary. Sibirsk. Mat. Zh. 21(6), 108–116 (1980)
Stephani, I.: Injectively \(A\)-compact operators, generalized inner entropy numbers, and Gel’fand numbers. Math. Nachr. 133, 247–272 (1987)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Allan Pinkus.
This research was partly supported by the Grant P201-13-14743S of the Grant Agency of the Czech Republic, by the Grant SVV-2017-260455 of the Charles University, and by the Neuron Fund for Support of Science.
Rights and permissions
About this article
Cite this article
Lang, J., Musil, V. Strict s-Numbers of Non-compact Sobolev Embeddings into Continuous Functions. Constr Approx 50, 271–291 (2019). https://doi.org/10.1007/s00365-018-9448-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00365-018-9448-0