Theorem 11.47.11278AD9E56CEA

Proof. The left tangents of $A$ lie below $A-M$ on $[a,b]$ once their contact point is less than a common $a_0<a$. They are minorants of every $A+F$, so all residuals vanish on the same neighborhood of $o$. The sandwich and slope bound give uniform real Hessian bounds on any fixed finite annulus, since $L_2\leq \| A''\| _\infty +M$. It remains to control infinity, where the right contact point need not stay bounded.

Put $B=\max (b,0)+1$. For $x>B$, the obstacle is $A$. At contact $C_ F'=A'$. On a free interval with finite right contact endpoint $d$, its constant slope is $A'(d)\geq A'(x)$; an unbounded right free interval has slope one. Thus

The locally Lipschitz functions $C_ F',A'$ agree on the contact set, so their derivatives agree almost everywhere there. Indeed, a differentiable Lipschitz function has derivative zero at almost every density point of its zero set. On the free set $C_ F''=0$. The reference formula consequently gives

In a chart $Z_ j\ne 0$, $j\geq 1$, use $\zeta =Z_0/Z_ j$ and $y=(Z_ i/Z_ j)_{i\geq 1,i\ne j}$. Then

On fixed relatively compact chart pieces, $|\nabla R|\leq c\sqrt R$ and $|\nabla ^2R|\leq c$. Write the envelope as $\mathcal E_ F(R)=C_ F(-\log R)+\log R-\log (1+R)$ for $R>0$. Its derivatives are

The tail bounds give $|(\mathcal E_ F)_ R|\leq 2$ and $|(\mathcal E_ F)_{RR}|\leq c_\alpha /R+1$. The real chain rule therefore bounds its Hessian uniformly, and its gradient tends to zero as $\zeta \to 0$. Also $L_ F=\lim (C_ F-x)\in [-M,0]$, and integrating $d_ F\leq \mathrm e^{-x}$ gives $|C_ F(x)-x-L_ F|\leq \mathrm e^{-x}$. Hence $\mathcal E_ F-L_ F=O(R)$, so the extended envelope is differentiable on the hyperplane with zero gradient. The Hessian bound makes the gradient Lipschitz off the hyperplane. Apply it along straight coordinate segments and use the continuous zero gradient at a crossing; on segments contained in the hyperplane the gradient is zero. This proves a uniform $C^{1,1}$ extension. Subtract the fixed smooth $u_\alpha $ near infinity and join this estimate to the common pole neighborhood and a fixed annulus.

For the stability estimate put $Z=R_ F-R_ G$. By (11.47.9), $\| Z\| _\infty \leq \delta $, and its gradient has a uniform Lipschitz bound. On a smaller fixed chart, a real directional increment of length $h$ gives $|\partial _ e Z|\leq 2\delta /h+Kh$. Taking $h$ of order $\sqrt\delta $, or enlarging the constant when limited by the chart radius, yields $\| \nabla Z\| _\infty \leq c\sqrt\delta $. For points a distance $t$ apart,

Dividing by $t^\gamma $ and splitting at $t=\sqrt\delta $ gives the estimate in a finite atlas. When $\delta =0$, the value bound gives equality. This argument does not assert convergence in $C^{1,1}$. $\square$