Lemma 5.42.5A340B19CD0CC

Proof. Use the Chern averaging operators and the classical estimate stated before lemma 5.42.4, with $\alpha $ in place of $\theta $. For some $K_0,A>0$ and $\epsilon _0(t)\to 0$, put $\lambda (x,t)=t\partial _ t(R_ tu(x)+K_0t^2)$. The estimate with vanishing radius component gives

The function $\lambda $ is nonnegative, continuous for $t>0$, and increasing in $t$. For fixed $x$, the convex function $s\mapsto R_{\mathrm{e}^ s}u(x)+K_0\mathrm{e}^{2s}$ has the finite limit $u(x)$ as $s\to -\infty $. Its nonnegative derivative must therefore tend to zero, since a positive limit of the derivative would make the function tend to $-\infty $. Thus $\lambda (x,t)\to 0$ as $t\to 0$, and for any sequence $t_ j\downarrow 0$, Dini's theorem gives $\| \lambda (\cdot ,t_ j)\| _\infty \to 0$.

The averaging kernels have mass one and only sample points at distance at most $Ct$ from the base point. Uniform continuity of $u$ therefore gives $\| R_ tu-u\| _\infty \to 0$. Set $u_ j=R_{t_ j}u$, and let $\varepsilon _ j\to 0$ be a decreasing positive majorant of $A\| \lambda (\cdot ,t_ j)\| _\infty +\epsilon _0(t_ j)$. $\square$