Let $(M, g)$ be a non-compact smooth Riemannian manifold of dimension $n \ge 2$, and $G$ a subgroup of the isometry group of $(M,g)$, say with $G$ contained in the component of the identy.

Let $W^{1,2}_{G}(M)=\{f \in W^{1,2}(M)| \quad f\circ \phi =\phi \quad \forall \phi \in G\}$.

Is there any known result concerning the compactness of the Sobolev imbedding $W^{1,2}_{G}(M) \hookrightarrow L^p(M)$ for some subgroup $G$?