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 <ref name="Lieb">E. Lieb, On the lowest eigenvalue of the Laplacian for the intersection of two domains. Invent. Math. '''74''' (1983), 441–448.</ref>(Lemma 6),<ref name="BC">V. Benci, G. Cerami, Existence of positive solutions of the equation −Δu+a(x)u=u(<sup>N+2)/(N−2)</sup> in R<sup>N</sup>, J. Funct. Anal. '''88''' (1990), no. 1, 90–117.</ref>(Lemma 2.5),<ref name="Solimini">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.</ref>(Theorem 1), or by ad-hoc monikers such as ''vanishing lemma'' or ''inverse embedding''.<ref name="Tao">Terence Tao, A pseudoconformal compactification of the nonlinear Schrödinger equation and applications, New York J. Math. '''15''' (2009), 265–282.</ref>
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 <math>G</math> be a group of isometries on a normed vector space <math>X</math>. One says that a sequence <math>(x_k)\subset X</math> converges to <math>x\in X</math> <math>G</math>-weakly, if for every sequence <math>(g_k)\subset G</math>, the sequence <math>g_k(x_k-x)</math> is weakly convergent to zero.
A continuous embedding of two normed vector spaces, <math>X\hookrightarrow Y</math> is called ''cocompact'' relative to a group of isometries <math>G</math> on <math>X</math> if every <math>G</math>-weakly convergent sequence <math>(x_k)\subset X</math> is convergent in <math>Y</math>.<ref>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.</ref>
== An elementary example: cocompactness for <math>\ell^\infty\hookrightarrow\ell^\infty</math> == Embedding of the space <math>\ell^\infty(\mathbb Z)</math> into itself is cocompact relative to the group <math>G</math> of shifts <math>(x_n)\mapsto (x_{n-j}), j\in\mathbb Z</math>. Indeed, if <math>(x_n)^{(k)}</math>, <math>k=1,2,\dots</math>, is a sequence <math>G</math>-weakly convergent to zero, then <math>x_{n_k}^{(k)}\to 0</math> for any choice of <math>n_k</math>. In particular one may choose <math>n_k</math> such that <math>2|x_{n_k}^{(k)}|\ge \sup_n|x_n^{(k)}|=\|(x_n)^{(k)}\|_\infty</math>, which implies that <math>(x_{n})^{(k)}\to 0</math> in <math>\ell^\infty</math>.
== Some known embeddings that are cocompact but not compact == * <math>\ell^p(\mathbb Z)\hookrightarrow \ell^q(\mathbb Z)</math>, <math>q< p</math>, relative to the action of translations on <math>\mathbb Z</math>:<ref name="Jaffard">S. Jaffard, Analysis of the lack of compactness in the critical Sobolev embeddings. J. Funct. Anal. '''161''' (1999).</ref> <math>(x_n)\mapsto (x_{n-j}), j\in\mathbb Z</math>. * <math> H^{1,p}(\mathbb R^N)\hookrightarrow L^q(\mathbb R^N)</math>, <math>p<q<\frac{pN}{N-p}</math>, <math>N>p</math>, relative to the actions of translations on <math>\mathbb R^N</math>.<ref name="Lieb" /> * <math> \dot H^{1,p}(\mathbb R^N)\hookrightarrow L^\frac{pN}{N-p}(\mathbb R^N)</math>, <math>N>p</math>, relative to the product group of actions of dilations and translations on <math>\mathbb R^N</math>.<ref name="BC" /><ref name="Solimini" /><ref name="Jaffard"/> * Embeddings of Sobolev space in the Moser–Trudinger case into the corresponding Orlicz space.<ref name="AT">Adimurthi, C. Tintarev, On compactness in the Trudinger–Moser inequality, Annali SNS Pisa Cl. Sci. (5) '''Vol. XIII''' (2014), 1–18.</ref> * Embeddings of Besov and Triebel–Lizorkin spaces.<ref name="BCC">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.</ref> * Embeddings of Strichartz spaces.<ref name="Tao" />
==References== {{reflist}}
Category:Compactness (mathematics) Category:Convergence (mathematics) Category:Functional analysis Category:General topology Category:Nonlinear functional analysis Category:Normed spaces