We apply these estimates to smooth approximations of the boundary values of $\varphi $ on a ball in $\Omega $. Recall that $\theta $, $\omega $, $c$ and $f$ are the data of theorem 15.18.1.
Proposition 15.18.21.59A5B60FAB74 Let $B\Subset \Omega $ be a coordinate ball, identified with $\{ |z|<r_0\} $, with closure in a larger coordinate chart, and put
\[ F=\frac{cf\, \omega ^ n}{\beta ^ n}. \]
There are $a>0$, $b,C<\infty $, and $u_ j\in C^\infty (\overline B)$ such that
9F54A2578D16
15.18.22
\begin{equation} \theta +\mathrm{dd}^{\mathrm{c}}u_ j>0, \qquad (\theta +\mathrm{dd}^{\mathrm{c}}u_ j)^ n=F\beta ^ n, \end{equation}
and
082B0A6218F2
15.18.23
\begin{equation} \lVert u_ j-\varphi \rVert _{L^\infty (B)}\leq Cj^{-a}, \qquad \sup _{\overline B}\operatorname {tr}_\beta (\theta +\mathrm{dd}^{\mathrm{c}}u_ j)\leq Cj^ b. \end{equation}
Proof.
Since $\varphi $ is Hölder continuous near $\overline B$, mollification at scale $1/j$ gives smooth boundary approximations $g_ j$ with
\[ \lVert g_ j-\varphi \rVert _{L^\infty (\partial B)}\leq Cj^{-a}, \qquad \lVert g_ j\rVert _{C^4(\partial B)}\leq Cj^4. \]
Choose a fixed quadratic $q$ with $\eta =\theta +\mathrm{dd}^{\mathrm{c}}q>0$ near $\overline B$. Extend $g_ j$ smoothly to $G_ j$. For a sufficiently large, polynomially bounded $A_ j$, the function
\[ \underline w_ j=G_ j-q+A_ j(|z|^2-r_0^2) \]
has boundary value $g_ j-q$ and satisfies
\[ \eta +\mathrm{dd}^{\mathrm{c}}\underline w_ j =\theta +\mathrm{dd}^{\mathrm{c}}G_ j+A_ j\beta >0, \qquad (\eta +\mathrm{dd}^{\mathrm{c}}\underline w_ j)^ n\geq F\beta ^ n. \]
Apply the smooth Dirichlet theorem recalled at the beginning of this subsection, with density $H=F\beta ^ n/\eta ^ n$ to obtain a solution $w_ j$ for the Hermitian background $\eta $; put $u_ j=w_ j+q$. Comparing $u_ j$ with $\varphi \pm \lVert g_ j-\varphi \rVert _\infty $ by proposition 15.18.3 proves the first bound in (15.18.23). The functions $u_ j$ are therefore uniformly bounded. Proposition 15.18.14 and the polynomial $C^4$ boundary norm give the second bound.
$\square$
Pome
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