Sobolev embedding on bounded extension domains for $p<n$

Given: The Axiom of Choice; $n\ge 2$; a bounded extension domain $\mathrm{\Omega }$ with a bounded extension operator $E$ and operator norm $\mathrm{\parallel }E\mathrm{\parallel }$ (Sobolev extension domains and extension operators); $1\le p<n$; $p\le q\le p^{\ast }$; a field $\mathbb{K}$.

For $1<p<n$, the whole-space Gagliardo-Nirenberg-Sobolev inequality holds (The Gagliardo-Nirenberg-Sobolev inequality for $1<p<n$). For $p=1$, compactly supported smooth density (Compactly supported smooth functions are dense in W^{k,p}(R^n)) extends The p=1 Gagliardo-Nirenberg-Sobolev inequality to $W^{1,1}({\mathbb{R}}^n)$: the smooth estimate on differences gives an $L^{n\mathrm{/}(n-1)}$ Cauchy sequence, completeness gives its limit, and successive almost-everywhere subsequences in that space and in $L^1$ identify the limit with the Sobolev class (Riesz-Fischer completeness of $L^p$ for $1\le p\le \mathrm{\infty }$, Complex Lp completeness and almost-everywhere subsequences). Thus $\mathrm{\parallel }F{\mathrm{\parallel }}_{p^{\ast }}\le C(n,p)\mathrm{\parallel }DF{\mathrm{\parallel }}_p$ for every $1\le p<n$ (The Sobolev conjugate exponent and the scaling identity).

The transfer corollary: if $N$ is a whole-space functional with $N_{\mathrm{\Omega }}(F{\mathrm{\mid }}_{\mathrm{\Omega }})\le N(F)$ and $N(F)\le C\mathrm{\parallel }F{\mathrm{\parallel }}_{W^{1,p}({\mathbb{R}}^n)}$, then $N_{\mathrm{\Omega }}(u)\le C\mathrm{\parallel }E\mathrm{\parallel }\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{1,p}(\mathrm{\Omega })}$ for all $u$ in $W^{1,p}(\mathrm{\Omega })$ (Whole-space inequalities transfer through a Sobolev extension).

Holder's inequality gives the $L^q$ interpolation bound $\mathrm{\parallel }v{\mathrm{\parallel }}_{L^q(\mathrm{\Omega })}\le \mathrm{\parallel }v{\mathrm{\parallel }}_{L^p(\mathrm{\Omega })}^{\theta }\mathrm{\parallel }v{\mathrm{\parallel }}_{L^{p^{\ast }}(\mathrm{\Omega })}^{1-\theta }$ whenever $1\le p\le q\le p^{\ast }<\mathrm{\infty }$ and $\frac{1}{q}=\frac{\theta }{p}+\frac{1-\theta }{p^{\ast }}$, on a finite measure set (Holder's inequality for integrals, including the endpoint cases).

$W^{1,p}$ consists of $L^p$ classes with weak gradient in $L^p$, and $\mathrm{\parallel }v{\mathrm{\parallel }}_{L^p(\mathrm{\Omega })}\le \mathrm{\parallel }v{\mathrm{\parallel }}_{W^{1,p}(\mathrm{\Omega })}$ (Integer-order Sobolev spaces and their norms, The space $L^p(\mu )$ as the quotient by null functions); every bounded $C^k$ domain supplies an admissible extension operator through the extension theorem (Bounded C^k domains admit integer-order Sobolev extension).