 Radial Poisson gradient estimate

For a radial Dirichlet solution on a fixed three-dimensional shell $a<r<b$, $(r^2{U}^{\prime }{)}^{\prime }=r^{2}F$. Thus $r^2{U}^{\prime }=C_0+{\int }_a^r{s}^{2}F(s)ds$, and $U(a)=U(b)=0$ bounds $|C_0|$ by the supremum of that integral. Cauchy-Schwarz inequality in the radial volume measure bounds the integral by $C\Vert f{\Vert }_2$. Since $r\ge a>0$, the displayed gradient bound follows. The estimate exploits radiality and does not extend to arbitrary forcing in three dimensions.