  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.
