The graph second form and electrovacuum Jacobi potential have weighted order four

Under [A1], the hypotheses match [F1] and [F2], giving $Du\in C_{-7\mathrm{/}4}^{4,\alpha }$, $D^{2}u\in C_{-11\mathrm{/}4}^{3,\alpha }$, a controlled induced inverse, and bounded normal components with four weighted Hölder derivatives. Apply [F7] to the metric bounds and the inverse bounds in [F6]. They give metric and inverse errors in $C_{-1}^{4,\alpha }$ and mixed components in $C_{-2}^{4,\alpha }$. Products add weights: on each fixed rescaled annulus, use the Leibniz rule and $[ab{]}_{\alpha }\le \mathrm{\parallel }a{\mathrm{\parallel }}_{\mathrm{\infty }}[b{]}_{\alpha }+\mathrm{\parallel }b{\mathrm{\parallel }}_{\mathrm{\infty }}[a{]}_{\alpha }$ for each differentiated term. The Christoffel formula ${\mathrm{\Gamma }}_{bc}^a=\frac{1}{2}{g}^{ad}({\mathrm{\partial }}_b{g}_{dc}+{\mathrm{\partial }}_c{g}_{db}-{\mathrm{\partial }}_d{g}_{bc})$, obtained from metric compatibility and torsion freedom, therefore gives ${\mathrm{\Gamma }}_{bc}^a\in C_{-2}^{2,\alpha }$. Stationarity further gives ${\mathrm{\Gamma }}_{ij}^0=\frac{1}{2}{g}^{00}({\mathrm{\partial }}_i{g}_{0j}+{\mathrm{\partial }}_j{g}_{0i})+\frac{1}{2}{g}^{0\mathrm{\ell }}({\mathrm{\partial }}_i{g}_{\mathrm{\ell }j}+{\mathrm{\partial }}_j{g}_{\mathrm{\ell }i}-{\mathrm{\partial }}_{\mathrm{\ell }}{g}_{ij})\in C_{-3}^{2,\alpha }$. In particular the zero-slope term has stronger decay than a general connection component.

Put $e_i={\mathrm{\partial }}_i+u_i{\mathrm{\partial }}_t$. Metric compatibility and $g(n,e_j)=0$ give $k_{ij}=g(n,{\mathrm{\nabla }}_{e_i}{e}_j)$ in the sign convention [F4]. By [F6], $g(n,\cdot )=-d(t-u)\mathrm{/}\sqrt{Q}$, and [F1] and [F6] imply that $Q$ tends to one and that $Q^{-1\mathrm{/}2}$ is bounded with controlled derivatives through order two. The second-derivative part of ${\mathrm{\nabla }}_{e_i}{e}_j$ contributes $-u_{ij}\mathrm{/}\sqrt{Q}$. Its connection part with no slope factor contributes $-{\mathrm{\Gamma }}_{ij}^0\mathrm{/}\sqrt{Q}$; every other term has at least one factor $Du$ multiplying a connection coefficient. Step 1.1 bounds these contributions respectively in $C_{-11\mathrm{/}4}^{2,\alpha }$, $C_{-3}^{2,\alpha }$ and $C_{-15\mathrm{/}4}^{2,\alpha }$ or faster. The product inequality in step 1.1 proves $k\in C_{-11\mathrm{/}4}^{2,\alpha }$ on the end.

Under [A1], the defining equation in [F3] is ${\mathrm{Ric}}_{ab}=2(F_{ac}{g}^{cd}{F}_{bd}-\frac{1}{4}{g}_{ab}{F}_{cd}{g}^{ce}{g}^{df}{F}_{ef})$. Its two field factors lie in $C_{-2}^{2,\alpha }$, while step 1.1 controls the metric and inverse factors, so the same differentiated product inequality gives Ricci components in $C_{-4}^{2,\alpha }$. The bounded normal components in [F1] preserve this weight upon contraction. Also $\mathrm{\mid }k{\mathrm{\mid }}_h^2=h^{ia}{h}^{jb}{k}_{ij}{k}_{ab}\in C_{-11\mathrm{/}2}^{2,\alpha }$ by [F1] and step 2.1. Since $r^{-11\mathrm{/}2}$ decays faster than $r^{-4}$, this implies the claimed order for $V$. Positive definiteness of $h$ makes $\mathrm{\mid }k{\mathrm{\mid }}_h^2$ nonnegative. The explicitly defined timelike-convergence condition applies to the unit timelike $n$, so it makes the other term nonnegative as well. Finally the assumed smooth spacelike extension over the compact core, including any compact boundary, makes $h^{-1}$, $k$, $n$ and the Ricci contraction smooth there; their norms on the fixed compact chart subsets are finite. This completes both global weighted claims and the conditional sign.