Lemma 14.5.7E5E1B9237922

Proof. Subtract $\inf _ Xv$, so that $0\leq v\leq M$, and put $K=\| \mathrm{d}v\| _{L^\infty (g)}$. Fix $p>2n$ and $0<\alpha <1-2n/p$. The fixed operator $\Delta _ g$ is uniformly elliptic on the underlying real manifold. The local $W^{2,p}$ estimates, patched by a finite atlas, and Morrey's inequality give

These are the standard linear elliptic estimates of [Chapter 9, GT01]; the constants depend on the fixed metric and on the chosen $p,\alpha $.

For $0<t\leq t_0$, with $t_0$ fixed by the atlas, integration along a coordinate segment of length $t$ gives

Indeed, the difference quotient is bounded by $2M/t$, and its difference from the derivative at the initial point is bounded by $C_ gt^\alpha \| v\| _{C^{1,\alpha }}$. A finite atlas with smaller charts controls every point and coordinate direction; equivalence with the $g$-norm gives the displayed estimate. Consequently,

Choose $t\leq t_0$ so small that $C_ g^2At^\alpha \leq 1/2$. This choice depends only on $g,A$, including when $A=0$. Absorbing the last term bounds $K$ in terms of $g,A,M$, and the assumed trace estimate then bounds $\| \Delta _ gv\| _\infty $. $\square$