Cocompact embedding

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

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

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

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