Sobolev-Poincare on bounded connected extension domains

Assume the Axiom of Choice (and hence Countable Choice and Dependent Choice). Let $n\ge 2$, $1<p<n$, $p^{\ast }=\frac{np}{n-p}$, let $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$, and let $\mathrm{\Omega }\subset {\mathbb{R}}^n$ be a nonempty bounded connected $W^{1,p}$-extension domain. There is $C=C(n,p,\mathrm{\Omega },\mathbb{K})$ such that every $u\in W^{1,p}(\mathrm{\Omega };\mathbb{K})$ satisfies $\mathrm{\parallel }u-u_{\mathrm{\Omega }}{\mathrm{\parallel }}_{L^{p^{\ast }}(\mathrm{\Omega })}\le C\mathrm{\parallel }Du{\mathrm{\parallel }}_{L^p(\mathrm{\Omega })},u_{\mathrm{\Omega }}=\mathrm{\mid }\mathrm{\Omega }{\mathrm{\mid }}^{-1}{\int }_{\mathrm{\Omega }}u\mathrm{.}$ The domain constant is not uniform over arbitrary extension domains.

Kinnunen's Theorem 3.47 is the mean-zero $L^{p^{\ast }}$ estimate, whose proof first establishes the mean-zero $L^p$ estimate on bounded connected extension domains, proved there by Rellich compactness; the local item cited as [F1] supplies it directly with the extension-cutoff-mollification and Arzela-Ascoli argument on this page. The step from the $L^p$ mean-zero estimate to the critical exponent is the extension-domain embedding, exactly as Kinnunen combines Theorem 3.47 with the Sobolev embedding. The constant depends on the extension operator through $C_E$; no uniformity over all extension domains is claimed, matching the statement.