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Zbl 0886.58003
Hebey, Emmanuel; Vaugon, Michel
Sobolev spaces in the presence of symmetries. (English)
[J] J. Math. Pures Appl., IX. Sér. 76, No.10, 859-881 (1997). ISSN 0021-7824

We prove that Sobolev embeddings can be improved in the presence of symmetries. This includes embeddings in higher $L^p$-spaces and compactness properties of these embeddings. While such phenomena have been observed in specific context by several authors, we treat here the case of arbitrary Riemannian manifolds (where, in particular, no global chart exist). On the one hand, it turns out that when dealing with compact manifolds, one just has to consider the minimum orbit dimension of the group acting on the manifold. On the other hand, and when dealing with non compact manifolds, one also has to consider the action of the group at infinity. Complete answers are given.
[E.Hebey (Paris)]

MSC 2000:: *58B20 Geometric structures on infinite-dimensional manifolds
46E35 Sobolev spaces and generalizations

Keywords: Sobolev embeddings; symmetries; embeddings in higher $L\sp p$-spaces; compactness; Riemannian manifolds

Cited in: Zbl 1126.46022 Zbl 1044.58032

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