Function spaces of Nikolskii type on compact manifold

Cristiana Bondioli

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1992)

  • Volume: 3, Issue: 3, page 185-194
  • ISSN: 1120-6330

Abstract

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Nikolskii spaces were defined by way of translations on R n and by way of coordinate maps on a differentiable manifold. In this paper we prove that, for functions with compact support in R n , we get an equivalent definition if we replace translations by all isometries of R n . This result seems to justify a definition of Nikolskii type function spaces on riemannian manifolds by means of a transitive group of isometries (provided that one exists). By approximation theorems, we prove that - for homogeneous spaces of compact connected Lie groups - our definition is equivalent to the classical one.

How to cite

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Bondioli, Cristiana. "Function spaces of Nikolskii type on compact manifold." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 3.3 (1992): 185-194. <http://eudml.org/doc/244171>.

@article{Bondioli1992,
abstract = {Nikolskii spaces were defined by way of translations on \( \mathbb\{R\}^\{n\} \) and by way of coordinate maps on a differentiable manifold. In this paper we prove that, for functions with compact support in \( \mathbb\{R\}^\{n\} \), we get an equivalent definition if we replace translations by all isometries of \( \mathbb\{R\}^\{n\} \). This result seems to justify a definition of Nikolskii type function spaces on riemannian manifolds by means of a transitive group of isometries (provided that one exists). By approximation theorems, we prove that - for homogeneous spaces of compact connected Lie groups - our definition is equivalent to the classical one.},
author = {Bondioli, Cristiana},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Nikolskii spaces; Isometry groups; Compact homogeneous spaces; Nikolskii space; space of absolutely integrable functions; differentiable manifold; Riemannian manifold; transitive group of isometries; homogeneous spaces of compact connected Lie groups},
language = {eng},
month = {9},
number = {3},
pages = {185-194},
publisher = {Accademia Nazionale dei Lincei},
title = {Function spaces of Nikolskii type on compact manifold},
url = {http://eudml.org/doc/244171},
volume = {3},
year = {1992},
}

TY - JOUR
AU - Bondioli, Cristiana
TI - Function spaces of Nikolskii type on compact manifold
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1992/9//
PB - Accademia Nazionale dei Lincei
VL - 3
IS - 3
SP - 185
EP - 194
AB - Nikolskii spaces were defined by way of translations on \( \mathbb{R}^{n} \) and by way of coordinate maps on a differentiable manifold. In this paper we prove that, for functions with compact support in \( \mathbb{R}^{n} \), we get an equivalent definition if we replace translations by all isometries of \( \mathbb{R}^{n} \). This result seems to justify a definition of Nikolskii type function spaces on riemannian manifolds by means of a transitive group of isometries (provided that one exists). By approximation theorems, we prove that - for homogeneous spaces of compact connected Lie groups - our definition is equivalent to the classical one.
LA - eng
KW - Nikolskii spaces; Isometry groups; Compact homogeneous spaces; Nikolskii space; space of absolutely integrable functions; differentiable manifold; Riemannian manifold; transitive group of isometries; homogeneous spaces of compact connected Lie groups
UR - http://eudml.org/doc/244171
ER -

References

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  1. BERNARDI, M. P. - BONDIOLI, G., Osservazioni ed esempi sugli spazi di Nikolskii. To appear. 
  2. BERGH, J. - LÖFSTRÖM, J., Interpolation Spaces. Springer Verlag, Berlin-New York1976. Zbl0344.46071MR482275
  3. HELGASON, S., Differential Geometry and Symmetric Spaces. Academic Press, New York-London1962. Zbl0111.18101MR145455
  4. KUFNER, A. - JOHN, O. - FUCIK, S., Function Spaces. Academia, Prague1977. Zblpre05948865MR482102
  5. MAGENES, E., On a Stefan problem on the boundary of a domain. Proceedings of the Intern. Meeting on Partial Differential Equations and related Problems (in honour to L. Nirenberg) (Trento, 3- 8/9/1990), to appear. Zbl0803.35170MR1190942
  6. MAGENES, E. - VERDI, C. - VISINTIN, A., Some theoretical and numerical results on the two-phase Stefan problem. SIAM J. Numer. Anal., 26, 1989, 1425-1438. Zbl0738.65092MR1025097DOI10.1137/0726083
  7. MATSUSHIMA, Y., Differentiable Manifolds. Marcel Dekker Inc., New York1972. Zbl0233.58001MR346831
  8. MEYER, Y., Ondelettes. Ondelettes et opérateurs I. Hermann, Paris1990. Zbl0694.41037MR1085487
  9. NIKOLSKII, S. M., Approximation of Functions of Several Variables and Imbedding Theorems. Springer Verlag, Berlin-New York1975. Zbl0185.37901MR374877
  10. RICCI, F. - STEIN, E. M., Harmonic analysis on nilpotent groups and singular integrals. II: Singular kernels supported on submanifolds. J. Funct. Anal., 78, 1988, 56-84. Zbl0645.42019MR937632DOI10.1016/0022-1236(88)90132-2
  11. STEIN, E. M., Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton, N.J., 1970. Zbl0207.13501MR290095
  12. TAIBLESON, M. H., On the theory of Lipschitz spaces of distributions on euclidean n -space. I. J. Math. Mech , 13, 1964, 407-479. Zbl0132.09402MR163159
  13. VARADARAJAN, V. S., An Introduction to Harmonic Analysis on Semisimple Lie Groups. Cambridge University Press, Cambridge, N.J., 1989. Zbl0924.22014MR1071183

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