Hostname: page-component-745bb68f8f-cphqk Total loading time: 0 Render date: 2025-01-27T23:03:03.876Z Has data issue: false hasContentIssue false
  • English
  • Français

Embeddings of Sobolev spaces of fractional order†

Published online by Cambridge University Press:  14 February 2012

J. S. Martins
J. S. Martins
Affiliation:
Department of Mathematics, University of Sussex and University of Coimbra, Portugal
Get access Rights & Permissions [Opens in a new window]

Synopsis

If Ω is a bounded domain in Rn satisfying certain conditions, Ωk denotes its intersection with a k-dimensional hyperplane, 1 ≦ k ≦ n, it is shown that the embedding of the Sobolev space Ws,p(Ω), s>0, into Lq(Ωk) is of type lm if for q<p<∞. The same result is obtained for the space of Bessel potentials Ls,p(Ω). Piecewise polynomial and Fourier approximations of functions and interpolation theorems areused.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 79 , Issue 1-2 , 1977 , pp. 1 - 24
Copyright
Copyright © Royal Society of Edinburgh 1977

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Adams, R. A.. Sobolev spaces (London: Academic Press, 1975).Google Scholar
2Aranda, P. J. and Cattaneo, E.. Classe P de l'injection de dans L2(Ω). CR. Acad. Sci. Paris Sér. A-B 274 (1974), A1292–A1295.Google Scholar
3Birman, M. S. and Solomjak, M. Z.. Approximation of the functions of the class by piecewise polynomial functions. Dokl Akad. Nauk SSSR 171 (1966) 1015–1018 and Sov. Math. Dokl. 7 (1966), 1573-1577.Google Scholar
4Birman, M. S. and Solomjak, M. Z.. Piecewise polynomial approximations of functions of class Mat. Sb. 115 (1967), 331–355.Google Scholar
5Clark, C.. The Hilbert-Schmidt property for embedding maps between Sobolev spaces. Canad. J. Math. 18 (1966), 1074–1084.CrossRefGoogle Scholar
6Edmunds, D. E. and Moscatelli, V. B.. Fourier approximation and embedding of Sobolev spaces. Dissertationes Math., to appear.Google Scholar
7Lions, J. L.. Sur les espaces d'interpolation: dualité. Math. Scand. 9 (1961), 147–177.Google Scholar
8Lions, J. L. and Magenes, E.. Problemi ai limiti non-omogenei. III. Ann. Scuola Norm. Sup. Pisa 15 (1961), 41–103.Google Scholar
9Martins, J. S.. Embeddings of Sobolev spaces on unbounded domains. Ann. Mat. Pura Appl., to appear.Google Scholar
10Maurin, K.. Metpdy prezestrzeni Hilberta. Monografie Mat. 36 (1959) and45 (1967).Google Scholar
11Nikol'skii, S. M.. On imbedding, continuation and approximation theorems for differentiable functions of several variables. Uspehi Mat Nauk 16, 5 (1961), 63-114 and Russian Math. Surveys 16, 5 (1961), 55-104.Google Scholar
12Lai, Phan The. Sur les operateurs compacts dans les espaces . C.R. Acad. Sci. Paris Sér. A-B 270 (1970), A1229–A1232.Google Scholar
13Pietsch, A.. Nuclear and locally convex spaces (Berlin: Springer, 1972).Google Scholar
14Sobolev, S. L.. Applications of Functional Analysis in Mathematical Physics. Izdat. Leningrad Gos. Univ. (1950) and Trans. Math. Monographs 7 (Providence: Amer. Math. Soc, 1963).Google Scholar
15Sobolev, S. L. and Nikol'skii, S. M.. Embedding theorems. Izdat. Akad. Nauk SSSR, Leningrad (1963) and Amer. Math. Soc. Transl. 87 (1970), 144–173.Google Scholar
16Stein, E. M.. The characterization of functions arising as potentials. Bull. Amer. Math. Soc. 68 (1962), 577–584.CrossRefGoogle Scholar
17Stein, E. M.. Singular integrals and differentiability properties of functions (Princeton: University Press, 1970.Google Scholar
18Triebel, H.. Uber die Verteilung der Approximationszahlen kompakter Operatoren in Sobolev-Besov-Räumeji. Invent. Math. 4 (1967), 275–293.CrossRefGoogle Scholar
19Triebel, H.. Uber die Approximationszahlen der Einbettungsoperatoren . Arch. Math. (Basel) 19 (1968), 305–312.CrossRefGoogle Scholar
20Uspensku, S. V.. An imbedding theorem for S. L. Sobolev's classes of fractional order Dokl. Akad. Nauk. SJSSR 130 (1960), 992–993 and Soviet Math. Dokl. 1 (1960), 132r133. vGoogle Scholar
21Uspenskh, S. V.. Properties of classes with a fractional derivative on differentiable manifolds. Dokl. Akad. Nauk SSSR 132 (1960), 60–62 and Soviet Math. Dokl. 1 (1960), 495–497.Google Scholar