A geometric criterion for compactness of invariant subspaces

Abstract

Abstract. Let M be a non-compact homogeneous Riemannian manifold, and let Ω be a compact subgroup of isometries of M . We show, under general conditions, that the Ω-invariant subspace AΩ of a normed vector space A → L q (M ) is compactly embedded into L q (M ) if and only if the group Ω has no orbits with a uniformly bounded diameter in a neighborhood of infinity. Mathematics Subject Classification (2010). Primary 46B50; Secondary 46E35, 46N20.

New geometric aspects of Moser–Trudinger inequalities on Riemannian manifolds: the non-compact case

“…In our case such approach fails. However, in order to prove Theorem 2.2, we shall combine the principle of symmetric criticality of Palais [44] with a recent characterization of compactness of invariant Sobolev spacesà la Lions (see [41]) under the action of isometries, see Skrzypczak and Tintarev [47]. As far as we know, the only result for V ≡ 1 in R n has been provided recently by doÓ, de Souza, de Medeiros and Severo [23] via a Lions-type concentration-compactness argument.…”

“…while the space W 1,n (M) is the completion of C ∞ 0 (M) with respect to the norm · 0,1 . In the sequel we adapt the main results from Skrzypczak and Tintarev [47] to our setting concerning the Sobolev spaces in the presence of group-symmetries; for a similar approach see also Hebey and Vaugon [29]. When (M, g) is a Hadamard manifold, the embedding W 1,n (M) ֒→ L p (M) is continuous for every p ∈ [n, ∞) (cf.…”

Compact Sobolev embeddings on non-compact manifolds via orbit expansions of isometry groups

“…Skrzypczak and Tintarev [41,44] identified general geometric conditions that are behind the compactness of Sobolev embeddings of the type W 1, p G (M) → L q (M) for certain ranges of p and q; their studies deeply depend on the curvature of the Riemannian manifold. In the light of their works, our purpose is twofold; namely, we provide an alternative characterization of the properties described by Skrzypczak and Tintarev [41,44] by using the expansion of geodesic balls and state the compact Sobolev embeddings of isometry-invariant Sobolev functions to Lebesgue spaces for the full admissible range of parameters. Given d ∈ N with d ≥ 2, we say that ( p, q) ∈ (1, ∞) × (1, ∞]…”

Looking for compactness in Sobolev spaces on noncompact metric spaces

“…Then it was shown by Berestycki-Lions [4,33], (see also Coleman-Glazer-Martin [9] and Strauss [37]) that W 1,p r (R n ) ֒→֒→ L q (R n ), where p < q < p * . The Berestycki-Lions type theorem has also been established on Riemannian manifolds (see Hebey-Vaugon [27] and Skrzypczak-Tintarev [35]). Namely, assume that G is a compact subgroup of the group of global isometries of the complete Riemannian manifold (M, g).…”