= Direct spherical-shell H2 estimate
{title2=$\|u\|_{H^2}\le C\|\Delta u\|_2$}
On a fixed shell $a<r<b$ with $a>0$, put $T=\partial_r^2+2r^{-1}\partial_r$ and $A=\Delta_{S^2}$. If $u$ is smooth and zero on both boundary spheres, direct radial and spherical <integration by parts> gives
$$
\|\Delta u\|_2^2=\|Tu\|_2^2+\|r^{-2}Au\|_2^2+2\int_a^b\!\int_{S^2}|\nabla_Su_r|^2-2\int_a^b\!\int_{S^2}r^{-2}|\nabla_Su|^2.
$$
The last term is controlled by the first-order energy estimate. The <spherical Hessian identity> controls angular second <derivatives>, and the radial/mixed <derivatives> follow from $T$ and $\nabla_Su_r$. The polar orthonormal-frame formulas for the Cartesian <Hessian matrix> then prove the estimate. Zero boundary values eliminate the radial boundary remainders because every angular <derivative> of their traces is zero.
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