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For a compact subgroup $G$ of the group of isometries acting on a Riemannianmanifold $M$ we investigate subspaces of Besov and Triebel-Lizorkin type whichare invariant with respect to the group action. Our main aim is to extend theclassical Strauss lemma under suitable assumptions on the Riemannian manifoldby proving that $G$-invariance of functions implies certain decay propertiesand better local smoothness. As an application we obtain inequalities ofCaffarelli-Kohn-Nirenberg type for $G$-invariant functions. Our resultsgeneralize those obtained by Skrzypczak. The main tool in our investigationsare atomic decompositions adapted to the $G$-action in combination with tracetheorems.

Symmetries on manifolds: Generalizations of the Radial Lemma of Strauss