Rellich compactness is strictly subcritical

The compactness statements of this page are strictly subcritical ; the companion page gives witnesses for the critical target exponents and for escape to infinity.

• (i) For $1\le p<n$ on a bounded domain the continuous embedding $W_0^{1,p}(\mathrm{\Omega })\hookrightarrow L^{p^{\ast }}(\mathrm{\Omega })$ is not compact, so the strict inequality $q<p^{\ast }$ in The Rellich--Kondrachov theorem for $1\le p<n$ on bounded extension domains and Subcritical compactness for $W_0^{1,p}$ on arbitrary bounded open sets cannot be replaced by $q=p^{\ast }$ (witness: a concentrating bump sequence ↗).
• (ii) On ${\mathbb{R}}^n$ boundedness in $W^{1,p}$ and translation control alone do not give compactness: the family of translates of one compactly supported bump satisfies the boundedness and translation hypotheses of The Fr'echet--Kolmogorov compactness criterion in $L^p({\mathbb{R}}^n)$ but not its tightness hypothesis, and has no strongly convergent subsequence (The tightness hypothesis of the Fr'echet--Kolmogorov criterion cannot be dropped ↗); the tightness hypothesis is therefore indispensable.
• (iii) In the Morrey range $p>n$ the endpoint H"older exponent $\beta =1-\frac{n}{p}$ is excluded from Morrey--Rellich compactness for $p>n$ (witness: rescaled H"older spikes ↗).
• (iv) At the critical source exponent $p=n$, Rellich--Kondrachov at the critical source exponent $p=n$ gives compactness into every finite $L^q$ and makes no $L^{\mathrm{\infty }}$ compactness claim. When $\mathrm{\Omega }$ is nonempty, the local log-log test function constructed in the proof of The critical Sobolev embedding into every finite $L^q$ belongs to $W^{1,n}(\mathrm{\Omega })$ and is essentially unbounded; this construction needs only an interior ball, independently of the extension hypotheses. For each $M>0$ its superlevel set $E_M$ has positive measure, so $\mathrm{\parallel }w{\mathrm{\parallel }}_{L^q}\ge M\mathrm{\mid }{E}_M{\mathrm{\mid }}^{1\mathrm{/}q}$ forces the best continuous embedding constants to tend to infinity as $q\to \mathrm{\infty }$. On the empty domain all spaces are zero.

For the $W^{1,p}$ compactness theorems on extension domains, boundedness and the extension hypothesis are part of the stated setting. The $W_0^{1,p}$ compactness results need only bounded openness and zero extension, with no boundary regularity. On ${\mathbb{R}}^n$, translations and dilations destroy compactness when the corresponding tail control is absent.