Sobolev spaces of integer order on compact homogeneous manifolds and invariant differential operators

Cristiana Bondioli

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1996)

  • Volume: 7, Issue: 4, page 219-233
  • ISSN: 1120-6330

Abstract

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Let M be a Riemannian manifold, which possesses a transitive Lie group G of isometries. We suppose that G , and therefore M , are compact and connected. We characterize the Sobolev spaces W p 1 ⁢ M 1 < p < + ∞ by means of the action of G on M . This characterization allows us to prove a regularity result for the solution of a second order differential equation on M by global techniques.

How to cite

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Bondioli, Cristiana. "Sobolev spaces of integer order on compact homogeneous manifolds and invariant differential operators." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 7.4 (1996): 219-233. <http://eudml.org/doc/244073>.

@article{Bondioli1996,
abstract = {Let \( M \) be a Riemannian manifold, which possesses a transitive Lie group \( G \) of isometries. We suppose that \( G \), and therefore \( M \), are compact and connected. We characterize the Sobolev spaces \( W\_\{p\}^\{1\} (M) \)\( ( 1 < p < + \infty ) \) by means of the action of \( G \) on \( M \). This characterization allows us to prove a regularity result for the solution of a second order differential equation on \( M \) by global techniques.},
author = {Bondioli, Cristiana},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Compact homogeneous manifolds; Sobolev spaces; Invariant differential operators; compact homogeneous manifolds; invariant differential operators; Lie-algebra; second-order differential operator; Laplace-Beltrami operator},
language = {eng},
month = {12},
number = {4},
pages = {219-233},
publisher = {Accademia Nazionale dei Lincei},
title = {Sobolev spaces of integer order on compact homogeneous manifolds and invariant differential operators},
url = {http://eudml.org/doc/244073},
volume = {7},
year = {1996},
}

TY - JOUR
AU - Bondioli, Cristiana
TI - Sobolev spaces of integer order on compact homogeneous manifolds and invariant differential operators
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1996/12//
PB - Accademia Nazionale dei Lincei
VL - 7
IS - 4
SP - 219
EP - 233
AB - Let \( M \) be a Riemannian manifold, which possesses a transitive Lie group \( G \) of isometries. We suppose that \( G \), and therefore \( M \), are compact and connected. We characterize the Sobolev spaces \( W_{p}^{1} (M) \)\( ( 1 < p < + \infty ) \) by means of the action of \( G \) on \( M \). This characterization allows us to prove a regularity result for the solution of a second order differential equation on \( M \) by global techniques.
LA - eng
KW - Compact homogeneous manifolds; Sobolev spaces; Invariant differential operators; compact homogeneous manifolds; invariant differential operators; Lie-algebra; second-order differential operator; Laplace-Beltrami operator
UR - http://eudml.org/doc/244073
ER -

References

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  1. ADAMS, R. A., Sobolev Spaces. Academic Press, New York1975. Zbl1098.46001MR450957
  2. AUBIN, T., Nonlinear Analysis on Manifolds. Monge-Ampère Equations. Springer-Verlag, New York-Heidelberg-Berlin1982. Zbl0512.53044MR681859DOI10.1007/978-1-4612-5734-9
  3. BERNARDI, M. P. - BONDIOLI, C., Function spaces W 1 , p , B ⁢ V , N λ , p characterized by the motions of R n . Annali Mat. Pura Appl., to appear. MR1625535DOI10.1007/BF01783471
  4. BIROLI, M. - MOSCO, U., Sobolev inequalities on homogeneous spaces. Potential Analysis, 4, 1995, 311-324. Zbl0833.46020MR1354886DOI10.1007/BF01053449
  5. BONDIOLI, C., Function spaces of Nikolskii type on compact manifolds. Rend. Mat. Acc. Lincei, s. 9, v. 3, 1992, 185-194. Zbl0772.46015MR1186914
  6. BONDIOLI, C., A regularity result of Nikolskii type for an evolution equation on compact homogeneous spaces. Math. Nachr., 166, 1994, 273-285. Zbl0837.35035MR1273338DOI10.1002/mana.19941660121
  7. BRÉZIS, H., Analyse fonctionnelle. Masson, Paris1983. Zbl0511.46001MR697382
  8. COIFMAN, R. R. - WEISS, G., Analyse harmonique sur certaines espaces homogènes. Lectures Notes in Mathematics, n. 242, Springer-Verlag, Berlin-Heidelberg-New York1971. Zbl0224.43006MR499948
  9. HELGASON, S., Groups and Geometric Analysis. Academic Press, New York1984. Zbl0543.58001MR754767
  10. LIZORKIN, P. I. - NIKOL'SKTJ, S. M., Approximations of functions on manifolds. In: H. J. SCHMEISSER - H. TRIEBEL (eds.), Function Spaces, Differential Operators and Nonlinear Analysis. Teubner, Stuttgart-Leipzig1993, 185-199. Zbl0798.41029MR1242583
  11. MATSUSHIMA, Y., Differentiable Manifolds. Marcel Dekkar Inc., New York1972. Zbl0233.58001MR346831
  12. MAZ'YA, V. G., Sobolev Spaces. Springer-Verlag, New York-Heidelberg-Berlin1985 (translation from Russian, Leningrad Univ., 1985). Zbl0692.46023MR817985
  13. NIRENBERG, L., Remarks on strongly elliptic partial differential equations. Comm. Pure Appl. Math., 8, 1955, 648-674. Zbl0067.07602MR75415
  14. ROTSCHILD, L. P. - STEIN, E. M., Hypoelliptic differential operators and nilpotent groups. Acta Math., 137, 1976, 247-320. Zbl0346.35030MR436223
  15. SKRZYPCZAK, L., Function spaces of Sobolev type on Riemannian symmetric manifolds. Forum Math., 3, 1991, 339-353. Zbl0738.46018MR1115951DOI10.1515/form.1991.3.339
  16. TRIEBEL, H., Theory of Function Spaces II. Birkhäuser, Basel-Boston-Berlin1992. Zbl0546.46027MR1163193DOI10.1007/978-3-0346-0419-2
  17. WALLACH, N. R., Harmonic Analysis on Homogeneous Spaces. Marcel Dekkar Inc., New York1973. Zbl0265.22022MR498996
  18. WARNER, F., Foundation of Differential Manifolds and Lie Groups. Academic Press, New York1971. 

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