On interpolation of cocompact imbeddings

Abstract

Cocompactness is a useful weaker counterpart of compactness in the study of imbeddings between function spaces. In this paper we prove that, under quite general conditions, cocompactness of imbeddings of Banach spaces persists under both real and complex interpolation. As an application, we obtain that subcritical continuous imbeddings of fractional Sobolev spaces and Besov spaces are cocompact relative to lattice shifts. We deduce this by interpolating the known cocompact imbeddings for classical Sobolev spaces ("vanishing" lemmas of Lieb and Lions). We also apply cocompactness to prove compactness of imbeddings of some radial subspaces and to show the existence of minimizers in some isoperimetric problems. Our research complements a range of previous results, and recalls that there is a natural conceptual framework for unifying them.

A geometric criterion for compactness of invariant subspaces

“…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 .…”

“…If X * is dense in A * , then weak cocompactness is trivially equivalent to cocompactness as defined in earlier work, e.g. [3].…”

“…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 .…”

Ground and bound state solutions for a Schrödinger system with linear and nonlinear couplings in $\mathbb{R}^N$

“…Proof. By continuity of Φ − Φ 0 with respect to the weak convergence it suffices to prove (4.5) for Φ = Φ 0 , which can be immediately obtained by iteration of the Brezis-Lieb lemma (see [8], Appendix B, for the scalar case). Since the map Φ ′ − Φ ′ 0 is continuous with respect to the weak convergence, the conclusion that (U (n) , V (n) ) is a critical point for respective functional is immediate.…”

Concentration Analysis and Cocompactness

“…In general, one would also expect that, given a common set of gauges, interpolation of two imbeddings, X 0 ֒→ Y 0 , and X 1 ֒→ Y 1 , results in a cocompact imbedding, if one of this imbeddings is cocompact. We refer to [20] where a more specific statement is proved, under additional conditions, for functional spaces of R N , which is then applied to verify that subcritical imbeddings of Besov spaces B s p,q (R N ) ֒→ B s1 p1,q1 (R N ), q 1 ≥ q, 0 < 1 p − 1 p1 < s−s1 N , s 1 ≥ 0.…”