Abstract
Let M be a non-compact homogeneous Riemannian manifold, and let Ω be a compact subgroup of isometries of M. We show, under general conditions, that the Ω-invariant subspace A Ω of a normed vector space \({A\hookrightarrow L^q(M)}\) is compactly embedded into L q(M) if and only if the group Ω has no orbits with a uniformly bounded diameter in a neighborhood of infinity.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Brézis H., Lieb E.: A relation between pointwise convergence of functions and convergence of functionals. Proc. Amer. Math. Soc. 88, 486–490 (1983)
Coleman S., Glazer V., Martin A.: Action minima among solutions to a class of euclidean scalar field equations. Comm. in Math. Physics 58, 211–221 (1978)
M. Cwikel and K. Tintarev, Rev. Mat. Complut. (online first) doi:10.1007/s13163-011-0087-2.
M.P. do Carmo, Riemannian Geometry, Birkhauser, Boston, Basel, Berlin, 1992.
E. Hebey, Nonlinear analysis on manifolds: Sobolev spaces and inequalities, AMS, Providence, Rhode Island 1999.
Hebey E., Vaugon M.: Sobolev spaces in the presence of symmetries. J. Math. Pures. Appl. 76, 34–70 (1997)
Lieb E.: On the lowest eigenvalue of the Laplacian for the intersection of two domains. Invent. Math. 74, 441–448 (1983)
Lions P.-L.: Symétrie et compacité dans les espaces de Sobolev. J. Funct. Anal. 49, 315–334 (1982)
Lions P.-L.: The concentration-compactness principle in the calculus of variations. The locally compact case, part 1, Ann. Inst. H. Poincare. Analyse non linéaire 1, 109–1453 (1984)
Schindler I., Tintarev K.: An abstract version of the concentration compactness principle. Revista Matematica Complutense, 15, 417–436 (2002)
Sickel W., Skrzypczak L.: Radial subspaces of Besov and Lizorkin-Triebel classes: extended Strauss lemma and compactness of embeddings. J. Fourier Analysus Appl. 6, 639–662 (2000)
Skrzypczak L.: Rotation invariant subspaces of Besov and Tribel-Lizorkin space: compactness of embeddings, smoothness and decay of functions, Revista Mat. Iberoamericana 18, 267–299 (2002)
Skrzypczak L.: Heat extensions, optimal atomic decompositions and Sobolev embeddings in presence of symmetries on manifolds. Math. Z. 243, 245–773 (2003)
Strauss W.A.: Existence of solitary waves in higher dimensions. Comm. in Math. Physics 55, 149–162 (1977)
Strichartz R.S.: Analysis of the Laplacian on a complete Riemannian manifold. J. Func. Anal. 52, 48–79 (1983)
Tao T.: A pseudoconformal compactification of the nonlinear Schrödinger equation and applications. New York J. Math. 15, 265–282 (2009)
K. Tintarev and K.-H. Fieseler Concentration-compactness: functional-analytic grounds and applications, Imperial College Press 2007.
Open Access
This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Skrzypczak, L., Tintarev, C. A geometric criterion for compactness of invariant subspaces. Arch. Math. 101, 259–268 (2013). https://doi.org/10.1007/s00013-013-0554-8
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00013-013-0554-8