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Write $r=|x|$, $u(x)=\bar{u}(r)$, and let $g(r)$ be the radial profile of the radial derivative $\nabla u(x)\cdot x/|x|$, which may be assumed to be a radial function here. The spherical coordinate system givesSince $1\le r^2\le 4$, both $\bar{u}$ and $g$ belong to the Lp space $L^6(1,2)$. It remains to identify the weak derivative of the radial profile directly.

For $\varphi \in C_c^{\infty }(1,2)$, each component of $\varphi (r)x/r$ is a test function on $A$. Sum the three defining weak derivative identities, using the stated divergence formula:Given an arbitrary test function $\zeta \in C_c^{\infty }(1,2)$, choose $\varphi (r)=\zeta (r)/r^2$. Dividing by $4\pi$ gives $\int g\zeta =-\int \bar{u}{\zeta }^{\prime }$, exactly the definition of the weak derivative. We have provedThis radial Sobolev profile on an annulus estimate follows from the two weighted integrals above and equivalence of the usual Sobolev norms.

Let $0<a<b<\infty$ and $1\le p<\infty$. 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 $\varphi (r)x/r$ to obtain $\int g\varphi r^{n-1}=-\int \bar{u}(\varphi r^{n-1}{)}^{\prime }$. Substitute $\varphi =\zeta /r^{n-1}$ 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 $|\bar{u}(r)-\bar{u}(s)|\le |r-s{|}^{1-1/p}\Vert {\bar{u}}^{\prime }{\Vert }_p$. 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 $||x|-|y||\le |x-y|$, and the reverse follows by putting both points on the same ray. The restriction $a>0$ is essential for the unweighted profile bound.