PomeLemma 5.42.7.3AB7626C8759 If $u\in \mathrm{PSH}(X,\omega )\cap C^{0,\alpha }(X)$ with $0<\alpha \leq 1$, there are smooth $u_ j\in \mathrm{PSH}(X,\omega )$ tending uniformly to $u$, with a common $C^{0,\alpha }$ bound. If $\sup _ Xu=0$, the approximants can also be normalized by $\sup _ Xu_ j=0$.
5.42.6 Hölder reference approximation and weighted smoothing91A88ECF3F18
The Chern averages preserve a Hölder norm uniformly. Their small lower-curvature loss can therefore be absorbed by rescaling, without changing the limiting potential.
Proof. Use the Chern averages $R_ tu$ in lemma 5.42.5. For a sequence $t_ j\downarrow 0$ there are $\varepsilon _ j\downarrow 0$ such that
\[ \omega +\mathrm{dd}^{\mathrm{c}}R_{t_ j}u\geq -\varepsilon _ j\omega , \qquad R_{t_ j}u\longrightarrow u\quad \text{uniformly}. \]
We also have a uniform Hölder bound. In a coordinate chart with a smooth orthonormal tangent frame, the averaging formula is
\[ R_ tu(x)=\int u(F(x,t\xi ))\rho (|\xi |^2)\, \mathrm d\xi . \]
On a smaller chart, smoothness of $F$ bounds $\operatorname {dist}(F(x,t\xi ),F(y,t\xi ))$ by $C\operatorname {dist}(x,y)$, uniformly for small $t$ and $|\xi |\leq 1$. Integrating the Hölder inequality for $u$ gives the same bound for $R_ tu$. A finite cover and the common supremum bound handle pairs outside a common smaller chart.
Set $u_ j=R_{t_ j}u/(1+\varepsilon _ j)$. Then
\[ \omega +\mathrm{dd}^{\mathrm{c}}u_ j =\frac{\varepsilon _ j\omega +(\omega +\mathrm{dd}^{\mathrm{c}}R_{t_ j}u)}{1+\varepsilon _ j}\geq 0. \]
These functions converge uniformly with a common Hölder bound. Subtracting their suprema gives the asserted normalization when $\sup _ Xu=0$. $\square$
Lemma 5.42.8.AEDE3F4F53EC Let $u\in \mathrm{PSH}(X,\omega )\cap C^{0,\alpha }(X)$, $0<\alpha \leq 1$, and let $\mu $ be a positive measure with $\mu \leq A\omega _ u^ n$. For bounded $\varphi \in \mathrm{PSH}(X,\omega )$ with $-M\leq \varphi \leq 0$, the Chern averages satisfy
\[ \| R_ t\varphi -\varphi \| _{L^1(\mu )} \leq C t^{2\alpha /(\alpha +2n)} \]
for small $t>0$. The constant is uniform under fixed bounds for $A,M$ and $\| u\| _{C^{0,\alpha }}$.
Proof. First we check a common lower curvature bound for the averages. The base part of the classical Chern curvature estimate stated before lemma 5.42.4 gives
\[ \omega +\mathrm{dd}^{\mathrm{c}}R_ t\varphi \geq -(A_0\lambda (\cdot ,t)+\epsilon _0(t))\omega , \qquad \lambda (x,t)=t\partial _ t(R_ t\varphi (x)+K_0t^2). \]
Its constants are fixed under the stated bounded range. In the local kernel form used in lemma 5.42.2,
\[ k_ t(x,y)=t^{-2n}a\bigl (x,(y-x)/t,t\bigr ), \]
with uniformly bounded first derivatives and compact support in its rescaled second variable. Differentiating with respect to $t$ at fixed $x,y$ and changing variables shows that
\[ \int |t\partial _ t k_ t(x,y)|\, \mathrm dy\leq C. \]
Thus $|t\partial _ tR_ t\varphi |\leq CM$, so $\lambda $ is uniformly bounded and the displayed curvature estimate supplies one fixed lower bound $\mathrm{dd}^{\mathrm{c}}R_ t\varphi \geq -C\omega $. Both $R_ t\varphi $ and $\varphi $ also lie in $[-M,0]$.
Apply corollary 5.29.9 to this pair, whose common lower bound has just been established. By lemma 5.42.2, its smooth-volume $L^1$ error is at most $Ct^2$. Domination by $A\omega _ u^ n$ therefore gives
\[ \int |R_ t\varphi -\varphi |\, \mathrm d\mu \leq C\| R_ t\varphi -\varphi \| _{L^1(\mathrm dV)}^{\alpha /(\alpha +2n)} \leq Ct^{2\alpha /(\alpha +2n)}. \]
$\square$
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