Direct spherical-shell H2 estimate ($\Vert u{\Vert }_{H^2}\le C\Vert \Delta u{\Vert }_2$)

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 givesThe 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 ${\nabla }_S{u}_r$. 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.