15.18.49 Proof of the main theorem6E838824045A

of theorem 15.18.1. If $n=1$, the equation is the linear identity

Its right-hand side is smooth, so interior elliptic regularity gives $\varphi \in C^\infty (\Omega )$, and the original equation gives $\theta +\mathrm{dd}^{\mathrm{c}}\varphi =cf\, \omega >0$. We henceforth assume $n\geq 2$.

Let

This is closed in $\Omega $. Suppose $\mathcal S\neq \varnothing $. By lemma 5.40.7, the restriction of $\varphi -\rho $ to $\mathcal S$ attains its minimum $m$ at some $p\in \mathcal S$.

Choose nested coordinate balls

centered at $p$. On $B_0$, proposition 15.18.21 gives smooth exact Dirichlet solutions $u_ j$ converging uniformly to $\varphi $ at a polynomial rate and having polynomial global trace growth.

On the open set $\Omega \setminus \mathcal S$ the function $\varphi $ is smooth, and $\theta +\mathrm{dd}^{\mathrm{c}}\varphi $ is semipositive with positive determinant, hence positive definite. Proposition 15.18.31 therefore gives a uniform trace bound for the $u_ j$ on every compact subset of $B_0\setminus \mathcal S$.

In coordinates centered at $p$, take $\kappa >0$ small and set

After shrinking $B$, the lower bound $\theta +\mathrm{dd}^{\mathrm{c}}\rho \geq \omega $ gives $\theta +\mathrm{dd}^{\mathrm{c}}q>0$ on $\overline B$. If $v=\varphi -q$, then the minimizing property of $p$ gives

Thus proposition 15.18.25 applies on $B$ with the closed set $\mathcal S\cap \overline B$: the approximation rate and global polynomial trace bound come from proposition 15.18.21, and the bound away from $\mathcal S$ comes from proposition 15.18.31. Its second assertion gives, for every $0<\alpha <1$, a neighborhood of $p$ on which the approximants have a uniform $C^{1,\alpha }$ bound.

Choose $\alpha >\alpha _*$ as in proposition 15.18.46, and apply that proposition on a ball centered at $p$ inside this neighborhood. It yields

on a still smaller neighborhood of $p$. Since $u_ j\to \varphi $ uniformly, $\mathrm{dd}^{\mathrm{c}}u_ j\to \mathrm{dd}^{\mathrm{c}}\varphi $ as currents. The trace bound passes to the limit, so $\operatorname {tr}_\beta (\theta +\mathrm{dd}^{\mathrm{c}}\varphi )$ has a bounded density near $p$. Proposition 15.18.5 now implies that $\varphi $ is smooth near $p$, contradicting $p\in \mathcal S$.

Hence $\mathcal S=\varnothing $ and $\varphi \in C^\infty (\Omega )$. The form $\theta +\mathrm{dd}^{\mathrm{c}}\varphi $ is smooth and semipositive there, while

Its nonnegative eigenvalues have positive product, so every eigenvalue is positive. Therefore $\theta +\mathrm{dd}^{\mathrm{c}}\varphi >0$ on $\Omega $. $\square$

Proof. The analytic complement of $\Omega $ has zero mass for both measures, so it suffices to know the equation in the bounded-potential sense on $\Omega $. Apply theorem 15.17.1 with density $cf$ to obtain local Hölder continuity. All hypotheses of theorem 15.18.1 are now satisfied. $\square$