On compact subsets of Sobolev spaces on manifolds

Abstract

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.