A geometric criterion for compactness of invariant subspaces

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 $${A\hookrightarrow L^q(M)}$$A↪Lq(M) is compactly embedded into Lq(M) if and only if the group Ω has no orbits with a uniformly bounded diameter in a neighborhood of infinity.


Introduction.
The problem studied in this paper has its origin in the compactness result of Strauss [14]: while the Sobolev embedding W 1,2 (R N ) → L q (R N ), q ∈ (2, 2N N −2 ), N > 2, is not compact, the subspace of radially symmetric functions of W 1,2 (R N ) is embedded into L q (R N ) compactly. Numerous generalizations of this result have been obtained for various domain and target spaces, as well as for different symmetry conditions, cf. [2,8,9,11,12]. In particular, the Sobolev spaces on Riemannian manifolds were studied by Hebey, Vaugon [5,6], and one of the authors [13]. A more abstract approach was developed by the second author and his collaborators [3,10,17] The purpose of this paper is to identify a general geometric condition that is necessary and sufficient for such a compactness phenomenon to occur in a setting when the original embedding is inherently non-compact. We consider an embedding of an abstract normed space A into the Lebesgue space of a non-compact complete Riemannian manifold M . All integrals in this paper are taken with respect to the Riemannian measure on M .
Throughout the paper we will use the following set of assumptions. (A) Let 1 < q < ∞, and let A → L q (M ) be a normed vector space. Let G be a subgroup of the group of isometries I(M ) of M that acts transitively The planar rotation action of SO(2) on R 3 is not coercive because for every t > 0 the set O t is a circular cylinder. The action of SO(3) on R 3 is coercive because for every t > 0 the set O t is the closed ball of radius t/2.
In order to formulate our main result, we need to define an important property of the space A.
An continuous embedding of a normed vector space A into a reflexive Banach space X is called weakly cocompact relative to a bounded set D of bounded linear operators acting on A if any bounded sequence u k ∈ A such that ∀g k ∈ D, g k u k 0 in X (which we write as u k D 0) converges to zero in the norm of X. Here denotes the weak convergence in X. In what follows we will also use the notation u k a.e. u in L q (M ) to indicate that the sequence u k converges weakly in L q and almost everywhere on M . Weak cocompactness of an embedding is a property related to, but generally weaker than compactness. If X * is dense in A * , then weak cocompactness is trivially equivalent to cocompactness as defined in earlier work, e.g. [3].
If the space A and the group G satisfy the assumption (A), then we can consider the weak cocompactness relative to the action of G, i.e. with respect to the set of isometries of A defined by the elements of the group G. If the group G is trivial, then cocompactness is trivially equivalent to compactness.
Cocompactness plays a crucial role in the existence of extrema in noncompact minimization problems, and is an underlying phenomenon of the concentration compactness principle introduced by Lions (e.g. [9]). In Section 4 we give a list of cocompact embeddings into L q , which of course includes Sobolev embeddings on homogeneous manifolds.
The conditions of the theorem are obviously satisfied by Sobolev spaces of R N and radial or suitable block-radial symmetries, cf. Section 3.
The paper is organized as follows. In Section 2 we prove the main theorem. In Section 3 we give a general geometric example of a coercive symmetry group, and in Section 4 we present examples of spaces A satisfying the conditions of the theorem. For convenience of the reader, we recall a covering lemma and an iterated Brezis-Lieb lemma in the last part of the paper.

Proof of main theorem.
We say that a sequence of points x n ∈ M converges to infinity if for any compact set K ⊂ M there is an integer N such that for all n ≥ N , x n ∈ K. Lemma 2.1. Assume that the group Ω is connected and acts coercively on M . Then for any sequence x k → ∞, there exists a sequence of elements ω Proof. Note that each orbit of Ω is a compact connected manifold. Let r > 0 and x ∈ M . By Lemma 5.1 there exist ω 1;r,x , . . . , ω m(r,x);r,x ∈ Ω such that Ωx is covered by the balls B 2r (ω 1;r,x x), . . . , B 2r (ω m(r,x);r,x x), while the balls The assertion follows from an elementary induction argument using the diagonal argument. We recall that the sequence η k ∈ I(M ) is called discrete if there exists a point x ∈ M such that η k (x) → ∞. Note that if the action of the group G is transitive and the property holds for some x, then it holds for every x in M . Moreover if η k is discrete, then also η −1 k is discrete. We are ready to prove Theorem 1.3.
Proof. Sufficiency: Let {u k } k∈N be an arbitrary bounded sequence in A Ω . We may assume without loss of generality that u k a.e. u in L q (M ). The embedding is weakly cocompact, therefore if u k has no subsequence convergent in L q (M ), then there is a sequence {η k } k∈N in G such that, on a renamed subsequence, we have The sequence η k −1 is necessarily discrete since if it is not discrete then η k −1 x converges for a renumbered subsequence and some x ∈ M . But η k −1 is a sequence of isometries of connected metric space, therefore convergence at one point implies, for a renumbered subsequence of η k −1 , convergence at any point of M . In consequence the sequence η k → η ∈ G in the compact-open topology of I(M ). Then for any But this contradicts the assumption that y = 0.
Let ω k → ω ∈ Ω. Then it follows from (2.1) that that converge in Ω as k → ∞ to, respectively, ω (1) , . . . , ω (J) , and such that d(ω k η k is discrete, and, consequently, the sequence for u with compact support and extends to the whole L q (M ) by density), the iterated Brezis-Lieb lemma, i.e. Lemma 5.1, applies and yields For a J sufficiently large (and a corresponding subsequence), we obtain a contradiction which proves the compactness of the embedding. Necessity: If Ω is not coercive, there exists R > 0 and a sequence x k → ∞ such that Ωx k ⊂ B R (x k ). By the assumption there exists a function ψ with M ψ > 0 supported in a ball B r (y) for some y ∈ M and r > 0. Let us replace x k with a renumbered subsequence such that the distance between any two terms in the sequence will be greater than 2(R + r). Let η k ∈ G be such that η k x k = y, and define where the Haar measure of Ω is normalized to the value 1. By the Young inequality, Note that the supports of the functions ψ k are disjoint, and therefore Furthermore, Consequently, we have a sequence, bounded in A and lacking a convergent subsequence in L q , and so the embedding A Ω → L q (M ) is not compact.

Remark 2.3.
• The reasoning in the proof of necessity extends with trivial changes to embeddings into any reflexive rearrangement-invariant space that is continuously embedded into L 1 loc (M ). Proof. We have A rad → A Ω .

Coercive groups.
In a more restrictive setting, we have the following necessary and sufficient condition for the action of a subgroup G ⊂ I(M ) to be coercive. Before we prove the proposition, we need a few auxiliary results. We will use the following statement, derived from [4, p. 218, Proposition 2.5]: The following statement is well known.
and consequently, the map ) is a diffeomorphism between orbits on S r (o) and S λr (o) for any r, λ > 0.
We are ready now to prove Proposition 3.1.
Proof. Necessity: By the Hadamard-Cartan theorem, M is diffeomorphic to R N under the exponential map at any given point of M . Assume that for some ρ > 0 there exists a fixed point x ρ ∈ S ρ (o). Then it follows from Lemma 3.3 that for any r > 0 the point is also a fixed point of Ω and d(x r , o) → ∞. So the group Ω is not coercive.
Sufficiency: Assume that there exists a constant C > 0 such that for a sequence R k → ∞, the geodesic sphere S R k (o) has a point x k such that diam Ωx k ≤ C. Now Proposition 3.2 implies is a subset of the unit sphere in T o M , and its diameter satisfies But then the group dΩ has a fixed point belonging to the unit sphere in T o M , so also Ω has a fixed point belonging to S 1 (o).
Example. In general, there exist coercive groups that have no fixed points, i.e. their orbits expand towards infinity and never shrink to a point. We give a simple example. Let M = S 1 × R n , n ≥ 2, be the Riemannian product manifold of the unit circle and the Euclidean space. Let Ω = S 1 × SO(n). Then Ω is a connected group of isometries acting on M . The action is given by the formulae (e iϕ , h)(e iψ , x) = (e i(ϕ+ψ) , h(x)), e iϕ , e iψ ∈ S 1 , h ∈ SO(n), and x ∈ R n . Every orbit of the group Ω has a form S 1 × S r (0) with some r ∈ [0, ∞). Therefore Ω is a compact subgroup of I(M ) that acts coercively on M and has no fixed point.  ∞), s ∈ [1, ∞) are cocompactly embedded into L q (M ), q ∈ (p, p * ), where p * = pN N −sp for sp < N and p * = ∞ for sp ≥ N , relative to I(M ). The cocompactness statement originates in the work of Lieb [7]. For the case of integer k, we refer the reader to [17]. For fractional s, the result follows from the continuity of Sobolev embeddings (see Strichartz [15]), the monotonicity of the Sobolev scale with respect to s, and the Hölder inequality. For M = R N , and 0 < s < 1, the cocompactness is verified in [3]. Thus, given the cocompactness of the embedding into L q (M ), Theorem 1.3 implies that, for any coercive compact subgroup Ω of I(M ), the Ω-invariant subspace of W s,p (M ) is compactly embedded into L q . In the case of R N , the hyperbolic space or a Carnot group, the coercivity of Ω means, by Proposition 3.1, that Ω has exactly one fixed point. In particular, in the case of R N or H N in the Poincaré ball coordinates, Ω may be SO(N ), or, more generally, consist of block-matrices ⎡ ⎢ ⎢ ⎣ For Besov spaces, the cocompactness of subcritical Jawerth's embeddings B s p,r (R N ) → L q (R N ), q ∈ (p, p * ), where p * = pN N −sp for sp < N and p * = ∞, was proved in [3]. Therefore, in this case the Ω-symmetric spaces of B s p,r (R N ) are cocompactly embedded into corresponding L q , if and only if Ω is coercive. When Ω is a group of block-symmetric matrices, this result was proved in [12], Corollary 3.
Let now M and Ω be as in Example 3. Then the Ω-invariant subspace W s,p Ω (M ) is compactly embedded into L q (M, μ) for any p < q < p * , where p * = p 1−sp if sp < 1 and p * = ∞ otherwise. This is immediate from the fact that the functions in W s,p Ω (M ) are in fact functions of one variable. This example, however, bears on the more general situation when the dimension of the orbit is involved in decreasing the effective local dimension of the manifold which may lead to a corresponding increase of the value of the critical Sobolev exponent. See more details in [6,13].
We would like also to mention the cocompactness of the Strichartz embeddings into L q for the nonlinear Schrödinger equation by Terence Tao [16] (Tao calls the cocompactness property "inverse embedding"). Similarly to the critical Sobolev inequality, the embedding of the radial subspace is not compact, as the loss of compactness may occur due to rescalings.