The maximal equation upgrades pointwise order-four graph decay to weighted order five

Under [A1], put $v(y)=u(Ry)$ and $w=v\mathrm{/}R$. The height and slope assumptions bound $w$ and keep $Dw$ in a fixed spacelike ball. [F1] and [F2] first give the realized coefficient bounds, before applying [F3]. The forcing for $v$ is $-Rs(y,0)=O(R^{-1})$ in $C^{2,\alpha }$. Thus $\mathrm{\parallel }v{\mathrm{\parallel }}_{C^{4,\alpha }}\le C(R^{-3\mathrm{/}4}+R^{-1})\le C{R}^{-3\mathrm{/}4}$ on a smaller fixed annulus. The conclusion includes Hölder control of fourth derivatives, not just their pointwise bounds.

One further differentiation is legitimate because the original solution is smooth. The realized coefficients now have uniform $C^{3,\alpha }$ bounds: differentiating $P_p(y,Dw)$ three times uses at most fourth derivatives of $w$, and differentiating ${\int }_0^1{s}_p(y,tDw)dt$ three times uses background spatial derivatives through order four. All finite slope derivatives are controlled by [F1]. The rescaled forcing has $C^{3,\alpha }$ norm $O(R^{-1})$. Differentiate the realized equation three times and, using the Countable Choice supplied by [A1], apply [F4] to each $D^{3}v$; every commutator contains coefficient derivatives through order three and solution derivatives through order four, already bounded by step 1.1. This bounds $v$ in $C^{5,\alpha }$ by $C{R}^{-3\mathrm{/}4}$ on a further smaller annulus.

The pointwise small slope and step 2.1 give, in the original coordinates, $a-\delta =O_{C^{2,\alpha }}(r^{-1})$, $b=O_{C^{2,\alpha }}(r^{-2})$ and $f=O_{C^{2,\alpha }}(r^{-3})$ by the explicit flux and zero-slope formulas. Here the quadratic slope correction is $O(r^{-7\mathrm{/}2})$ and does not weaken the metric error. Hence ${\mathrm{\Delta }}_{0}u=f-(a-\delta ):D^{2}u-b\cdot Du$ has weighted order $-3$ through two Hölder derivatives. Rescale back and use the fixed finite annular covers for the global end assertion.