The Rellich--Kondrachov theorem for $1\le p<n$ on bounded extension domains

Given: the Axiom of Choice, a bounded extension domain $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$, $1\le p<n$, $p^{\ast }=np\mathrm{/}(n-p)$, a target exponent $1\le q<p^{\ast }$, and a sequence $(u_j)$ with $M:={\sup }_j\mathrm{\parallel }{u}_j{\mathrm{\parallel }}_{W^{1,p}(\mathrm{\Omega })}<\mathrm{\infty }$.

$L^p$-compactness. Some subsequence converges in $L^p(\mathrm{\Omega })$. (Compactness of $W^{1,p}(\mathrm{\Omega })\hookrightarrow L^p(\mathrm{\Omega })$ on bounded extension domains, Sobolev extension domains and extension operators, Compactly embedded normed spaces)

Sobolev embedding on bounded extension domains. $\mathrm{\parallel }v{\mathrm{\parallel }}_{L^{p^{\ast }}(\mathrm{\Omega })}\le C\mathrm{\parallel }v{\mathrm{\parallel }}_{W^{1,p}(\mathrm{\Omega })}$ for all $v\in W^{1,p}(\mathrm{\Omega })$, $C=C(n,p,\mathrm{\Omega })$, hence $\mathrm{\parallel }{u}_j{\mathrm{\parallel }}_{L^{p^{\ast }}(\mathrm{\Omega })}\le CM$ and $\mathrm{\parallel }{u}_k-u_{\mathrm{\ell }}{\mathrm{\parallel }}_{L^{p^{\ast }}(\mathrm{\Omega })}\le 2CM$. (Sobolev embedding on bounded extension domains for $p<n$, The Sobolev conjugate exponent and the scaling identity, Integer-order Sobolev spaces and their norms)

For $p=1$, supply the endpoint separately: take the given extension $v=Eu\in W^{1,1}({\mathbb{R}}^n)$ and smooth compactly supported $v_j\to v$ in $W^{1,1}$ by Compactly supported smooth functions are dense in W^{k,p}(R^n). The endpoint The p=1 Gagliardo-Nirenberg-Sobolev inequality on differences makes $(v_j)$ Cauchy in $L^{n\mathrm{/}(n-1)}$; completeness and almost-everywhere subsequences identify this limit with $v$, since $v_j\to v$ in $L^1$. Passing to the limit gives $\mathrm{\parallel }v{\mathrm{\parallel }}_{n\mathrm{/}(n-1)}\le C\mathrm{\parallel }Dv{\mathrm{\parallel }}_1\le C\mathrm{\parallel }E\mathrm{\parallel }\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{1,1}}$, and restriction gives [F2]. (Riesz-Fischer completeness of $L^p$ for $1\le p\le \mathrm{\infty }$, Assuming Countable Choice, $L^p$-convergent sequences have almost-everywhere convergent subsequences)

Interpolation and H"older. For $p\le b<p^{\ast }$, $\mathrm{\parallel }g{\mathrm{\parallel }}_b\le \mathrm{\parallel }g{\mathrm{\parallel }}_p^{\theta }\mathrm{\parallel }g{\mathrm{\parallel }}_{p^{\ast }}^{1-\theta }$ with $1\mathrm{/}b=\theta \mathrm{/}p+(1-\theta )\mathrm{/}{p}^{\ast }$; for $b\le p$, $\mathrm{\parallel }g{\mathrm{\parallel }}_{L^b(\mathrm{\Omega })}\le \mathrm{\mid }\mathrm{\Omega }{\mathrm{\mid }}^{1\mathrm{/}b-1\mathrm{/}p}\mathrm{\parallel }g{\mathrm{\parallel }}_{L^p(\mathrm{\Omega })}$. (Lyapunov interpolation inequality for $L^p$ norms, Holder's inequality for integrals, including the endpoint cases, The space $L^p(\mu )$ as the quotient by null functions)

Under Countable Choice, each $L^q(\mathrm{\Omega })$, $1\le q<\mathrm{\infty }$, is complete. (Riesz-Fischer completeness of $L^p$ for $1\le p\le \mathrm{\infty }$)