Symmetries on manifolds: generalizations of the radial lemma of Strauss. (English) Zbl 1428.46023
Summary: For a compact subgroup \(G\) of the group of isometries acting on a Riemannian manifold \(M\) we investigate subspaces of Besov and Triebel-Lizorkin type which are invariant with respect to the group action. Our main aim is to extend the classical Strauss lemma under suitable assumptions on the Riemannian manifold by proving that \(G\)-invariance of functions implies certain decay properties and better local smoothness. As an application, we obtain inequalities of Caffarelli-Kohn-Nirenberg type for \(G\)-invariant functions. Our results generalize those obtained in [L. Skrzypczak, Rev. Mat. Iberoam. 18, No. 2, 267–299 (2002; Zbl 1036.46028)]. The main tool in our investigations are atomic decompositions adapted to the \(G\)-action in combination with trace theorems.
MSC:
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 53C20 | Global Riemannian geometry, including pinching |
Keywords:
symmetries on manifolds; Besov space; Triebel-Lizorkin space; Sobolev spaces; atomic decompositionsCitations:
Zbl 1036.46028References:
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