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 $\sup _ Xu_ j=0$.
Lemma 5.42.73AB7626C8759
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 $\sup _ Xu=0$. $\square$
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