has the embedding $$L^{p}_1(X,\mu) \hookrightarrow L^q(X,\mu)$$ whenever $p \leq q \leq p^{\ast} := n p / (n-p)$, where $1 \leq p < n$. If $q < p^{\ast}$, the embedding is compact, and one writes $$\label{comp-emb-1}
L^{p}_1(X,\mu) \hookrightarrow \hookrightarrow L^q(X,\mu) ,$$ see [@Adams; @aubin; @hebey]. The non-compactness of the embedding in the limit case $q = p^{\ast}$ is a phenomena that arises from sequences of transformations that induce substantial changes in the functions, transformations that nonetheless leave the norm of functions unchanged. With such information, it is tempting to look for subspaces of $L^{p}_1(X,\mu)$ whose elements are invariant under an appropriate subgroup of $\text{Diff}(X)$, and see if the compact embedding (\[comp-emb-1\]), when restricted to these subspaces, can be extended to higher values of $q$: let $H$ be a subgroup of $\text{Diff}(X)$, and denote by $L^{p}_{1,H}(X,\mu)$ the subspace of $L^{p}_1(X,\mu)$ made up of $H$-invariant functions.\
The best result in this context is due to E. Hebey and M. Vaugon, who consider $H$ as being a subgroup of $\text{Isom}(X,g)$, the group of isometries of $(X,g)$:

(Hebey-Vaugon [@hebey-vaugon], also Theorem 9.1 in [@hebey]) \[heb-vaug\] Suppose $(X,g)$ is a compact Riemannian $n$-manifold, and $H$ is a compact subgroup of $\text{Isom}(X,g)$. If $H(x)$ denotes the orbit 