# Cocompact embedding

In mathematics, **cocompact embeddings** are embeddings of normed vector spaces possessing a certain property similar to but weaker than compactness. Cocompactness has been in use in mathematical analysis since the 1980s, without being referred to by any name ^{[1]}(Lemma 6),^{[2]}(Lemma 2.5),^{[3]}(Theorem 1), or by ad-hoc monikers such as *vanishing lemma* or *inverse embedding*.^{[4]}

Cocompactness property allows to verify convergence of sequences, based on translational or scaling invariance in the problem, and is usually considered in the context of Sobolev spaces. The term *cocompact embedding* is inspired by the notion of cocompact topological space.

## Definitions

Let be a group of isometries on a normed vector space . One says that a sequence converges to -weakly, if for every sequence , the sequence is weakly convergent to zero.

A continuous embedding of two normed vector spaces, is called *cocompact* relative to a group of isometries on if every -weakly convergent sequence is convergent in .^{[5]}

## An elementary example: cocompactness for

Embedding of the space into itself is cocompact relative to the group of shifts . Indeed, if , , is a sequence -weakly convergent to zero, then for any choice of . In particular one may choose such that , which implies that in .

## Some known embeddings that are cocompact but not compact

- , , relative to the action of translations on :
^{[6]}. - , , , relative to the actions of translations on .
^{[1]} - , , relative to the product group of actions of dilations and translations on .
^{[2]}^{[3]}^{[6]} - Embeddings of Sobolev space in the Moser–Trudinger case into the corresponding Orlicz space.
^{[7]} - Embeddings of Besov and Triebel–Lizorkin spaces.
^{[8]} - Embeddings of Strichartz spaces.
^{[4]}

## References

- 1 2 E. Lieb, On the lowest eigenvalue of the Laplacian for the intersection of two domains. Invent. Math.
**74**(1983), 441–448. - 1 2 V. Benci, G. Cerami, Existence of positive solutions of the equation −Δu+a(x)u=u(
^{N+2)/(N−2)}in R^{N}, J. Funct. Anal.**88**(1990), no. 1, 90–117. - 1 2 S. Solimini, A note on compactness-type properties with respect to Lorentz norms of bounded subsets of a Sobolev space. Ann. Inst. H. Poincaré Anal. Non Linéaire
**12**(1995), 319–337. - 1 2 Terence Tao, A pseudoconformal compactification of the nonlinear Schrödinger equation and applications, New York J. Math.
**15**(2009), 265–282. - ↑ C. Tintarev, Concentration analysis and compactness, in: Adimuri, K. Sandeep, I. Schindler, C. Tintarev, editors, Concentration Analysis and Applications to PDE ICTS Workshop, Bangalore, January 2012, ISBN 978-3-0348-0372-4, Birkhäuser, Trends in Mathematics (2013), 117–141.
- 1 2 S. Jaffard, Analysis of the lack of compactness in the critical Sobolev embeddings. J. Funct. Anal.
**161**(1999). - ↑ Adimurthi, C. Tintarev, On compactness in the Trudinger–Moser inequality, Annali SNS Pisa Cl. Sci. (5)
**Vol. XIII**(2014), 1–18. - ↑ H. Bahouri, A. Cohen, G. Koch, A general wavelet-based
profile decomposition in the critical embedding of function spaces,
Confluentes Matematicae
**3**(2011), 387–411.