Cocompact embedding

Last updated March 20, 2019

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]

In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.

In mathematics, a normed vector space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of distance in the real world. A norm is a real-valued function defined on the vector space that has the following properties:

The zero vector, 0, has zero length; every other vector has a positive length.
Multiplying a vector by a positive number changes its length without changing its direction. Moreover,
The triangle inequality holds. That is, taking norms as distances, the distance from point A through B to C is never shorter than going directly from A to C, or the shortest distance between any two points is a straight line.

In mathematics, the notion of being compactly embedded expresses the idea that one set or space is "well contained" inside another. There are versions of this concept appropriate to general topology and functional analysis.

In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of L^p-norms of the function itself and its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, thus a Banach space. Intuitively, a Sobolev space is a space of functions with sufficiently many derivatives for some application domain, such as partial differential equations, and equipped with a norm that measures both the size and regularity of a function.

In mathematics, an action of a group G on a topological space X is cocompact if the quotient space X/G is a compact space. If X is locally compact, then an equivalent condition is that there is a compact subset K of X such that the image of K under the action of G covers X. It is sometimes referred to as mpact, a tongue-in-cheek reference to dual notions where prefixing with "co-" twice would "cancel out".

Definitions

Let $G$ be a group of isometries on a normed vector space $X$ . One says that a sequence $(x_{k})\subset X$ converges to $x\in X$ $G$ -weakly, if for every sequence $(g_{k})\subset G$ , the sequence $g_{k}(x_{k}-x)$ is weakly convergent to zero.

A continuous embedding of two normed vector spaces, $X\hookrightarrow Y$ is called cocompact relative to a group of isometries $G$ on $X$ if every $G$ -weakly convergent sequence $(x_{k})\subset X$ is convergent in $Y$ .^[5]

In mathematics, one normed vector space is said to be continuously embedded in another normed vector space if the inclusion function between them is continuous. In some sense, the two norms are "almost equivalent", even though they are not both defined on the same space. Several of the Sobolev embedding theorems are continuous embedding theorems.

An elementary example: cocompactness for $\ell ^{\infty }\hookrightarrow \ell ^{\infty }$

Embedding of the space $\ell ^{\infty }(\mathbb {Z} )$ into itself is cocompact relative to the group $G$ of shifts $(x_{n})\mapsto (x_{n-j}),j\in \mathbb {Z}$ . Indeed, if $(x_{n})^{(k)}$ , $k=1,2,\dots$ , is a sequence $G$ -weakly convergent to zero, then $x_{n_{k}}^{(k)}\to 0$ for any choice of $n_{k}$ . In particular one may choose $n_{k}$ such that $2|x_{n_{k}}^{(k)}|\geq \sup _{n}|x_{n}^{(k)}|=\|(x_{n})^{(k)}\|_{\infty }$ , which implies that $(x_{n})^{(k)}\to 0$ in $\ell ^{\infty }$ .

Some known embeddings that are cocompact but not compact

$\ell ^{p}(\mathbb {Z} )\hookrightarrow \ell ^{q}(\mathbb {Z} )$ , $q<p$ , relative to the action of translations on $\mathbb {Z}$ :^[6] $(x_{n})\mapsto (x_{n-j}),j\in \mathbb {Z}$ .
$H^{1,p}(\mathbb {R} ^{N})\hookrightarrow L^{q}(\mathbb {R} ^{N})$ , $p<q<{\frac {pN}{N-p}}$ , $N>p$ , relative to the actions of translations on $\mathbb {R} ^{N}$ .^[1]
${\dot {H}}^{1,p}(\mathbb {R} ^{N})\hookrightarrow L^{\frac {pN}{N-p}}(\mathbb {R} ^{N})$ , $N>p$ , relative to the product group of actions of dilations and translations on $\mathbb {R} ^{N}$ .^[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]

In mathematical analysis, Trudinger's theorem or the Trudinger inequality is a result of functional analysis on Sobolev spaces. It is named after Neil Trudinger.

In applied mathematics, Strichartz estimates are a family of inequalities for linear dispersive partial differential equations. These inequalities establish size and decay of solutions in mixed norm Lebesgue spaces. They were first noted by Robert Strichartz and arose out of contentions to the Fourier restriction problem.

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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.

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[Lieb-1] 1 2 E. Lieb, On the lowest eigenvalue of the Laplacian for the intersection of two domains. Invent. Math. 74 (1983), 441–448.

[BC-2] 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.

[Solimini-3] 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.

[Tao-4] 1 2 Terence Tao, A pseudoconformal compactification of the nonlinear Schrödinger equation and applications, New York J. Math. 15 (2009), 265–282.

[5] 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.

[Jaffard-6] 1 2 S. Jaffard, Analysis of the lack of compactness in the critical Sobolev embeddings. J. Funct. Anal. 161 (1999).

[AT-7] Adimurthi, C. Tintarev, On compactness in the Trudinger–Moser inequality, Annali SNS Pisa Cl. Sci. (5) Vol. XIII (2014), 1–18.

[BCC-8] 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.