It is common that a Sobolev space defined on $\mathbb{R}^m$ has a non-compact embedding into an $L^p$-space, but it has subspaces for which this embedding becomes compact. There are three well known cases of such subspaces, the Rellich compactness, for a subspace of functions on a bounded domain (or an unbounded domain, sufficiently thin at infinity), the Strauss compactness, for a subspace of radially symmetric functions in $\mathbb{R}^m$, and the weighted Sobolev spaces. Known generalizations of Strauss compactness include subspaces of functions with block-radial symmetry, subspaces of functions with certain symmetries on Riemannian manifolds, as well as similar subspaces of more general Besov and Triebel-Lizorkin spaces. Presence of symmetries can be interpreted in terms of the rising critical Sobolev exponent corresponding to the smaller effective dimension of the quotient space.

中文翻译：

---

关于流形上 Sobolev 空间的紧子集

定义在 $\mathbb{R}^m$ 上的 Sobolev 空间在 $L^p$-空间中具有非紧致嵌入是很常见的，但是它具有使该嵌入变得紧致的子空间。这种子空间有三种众所周知的情况，对于有界域（或无界域，在无穷远处足够薄）上的函数子空间的 Rellich 紧致性，对于 $\mathbb 中径向对称函数的子空间的 Strauss 紧致性{R}^m$ 和加权 Sobolev 空间。Strauss 紧致性的已知推广包括具有块径向对称性的函数子空间、黎曼流形上具有某些对称性的函数子空间，以及更一般的 Besov 和 Triebel-Lizorkin 空间的类似子空间。