= Radial Sobolev profile on an annulus
{title2=$\|\bar u\|_{W^{1,p}(a,b)}\leq C\|u\|_{W^{1,p}(\{a<|x|<b\})}$}
Let $0<a<b<\infty$ and $1\leq p<\infty$. A <radial function> in $W^{1,p}(\{a<|x|<b\})$ has a <radial profile> in $W^{1,p}(a,b)$; its first <weak derivative> is the profile of the <radial derivative>. To see the identity, sum the <weak derivative> tests against the components of $\phi(r)x/r$ to obtain $\int g\phi r^{n-1}=-\int\bar u(\phi r^{n-1})'$. Substitute $\phi=\zeta/r^{n-1}$ for arbitrary $\zeta\in C_c^\infty(a,b)$. The weighted <Lp norms> and $a>0$ give the displayed bound. Radiality of the <radial derivative> follows by applying a rotation to the <weak derivative> identity: if $u(Qx)=u(x)$, then $\nabla u(Qx)=Q\nabla u(x)$ <almost everywhere>, so the radial scalar product is invariant.
For $p>1$, the <one-dimensional Sobolev representative> obeys $|\bar u(r)-\bar u(s)|\leq|r-s|^{1-1/p}\|\bar u'\|_p$. The <reverse triangle inequality> transfers this to $u$, giving a <continuous embedding> of this radial <Sobolev space> into $C^{0,1-1/p}$ on the <annulus>. The two <Hölder seminorms> are equal: one inequality follows from $\bigl||x|-|y|\bigr|\leq|x-y|$, and the reverse follows by putting both points on the same ray. The restriction $a>0$ is essential for the unweighted profile bound.
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