On compact subsets of Sobolev spaces on manifolds

2020/03/13 by Skrzypczak, Leszek, Tintarev, Cyril
#Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2003.06456

Abstract

It is common that a Sobolev space defined on 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 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.

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