Question

It is known that, for $n \ge 3, 2 < p< 2^*$, the imbedding $H^1(\mathbb{R}^n) \hookrightarrow L^p(\mathbb{R}^n)$ is not compact. However Let $G=O(n_1) \times O(n_2)\times\cdots\times O(n_k)$, with $n_1+n_2+\cdots+n_k=n, n_i \ge 2$, and $k \ge 1$. Define an action of $G$ on $H^1(\mathbb{R}^n)$ by $g.u=u\circ g^{-1}$, and denote by $H^1_G(\mathbb{R}^n)$ the subspace $H^1_r(\mathbb{R}^n)$ of $H^1(\mathbb{R}^n)$ which consists of radial functionsthe fixed points of that action, i.e. functions of $|x|$, is compactly imbedded into g.u=u$ for all $L^p(\mathbb{R}^n)$.The g \in G$. Then the imbedding $H^1_G(\mathbb{R}^n) \hookrightarrow L^p(\mathbb{R}^n)$ is compact. The question is whether there exists a space $E \subsetneq H^1(\mathbb{R}^n)$, with $H^1_r(\mathbb{R}^n) H^1_G(\mathbb{R}^n) \subsetneq E$or $H^1_r(\mathbb{R}^n) \cap E=\emptyset$, that is compactly imbedded into $L^p(\mathbb{R}^n)$ ?