Lemma 8.9.1C35233382EB9

Proof. Choose smooth strictly admissible $u_ j\downarrow u$ by 8.6.6, and put $\sigma _ j=\omega +\mathrm{dd}^{\mathrm{c}}u_ j$. After discarding finitely many terms, $\| u_ j\| _\infty \leq \| u\| _\infty +1$. Indeed, the increasing open sets $\{ u_ j<\sup _ Xu+1\} $ cover $X$, and compactness gives a single index whose open set is all of $X$; the lower bound follows from $u_ j\geq u$.

The closed $m$-cone lies in the closed $2$-cone. In an $\omega $-unitary frame, its eigenvalues satisfy

Thus every coefficient of $\sigma _ j$ is bounded by its trace. The degree-one Chern–Levine–Nirenberg estimate in 8.7.1, applied to these smooth functions, bounds the integral of that trace on $K$ by $C(1+\| u\| _\infty )$. Subtracting the fixed smooth $\omega $ gives the same bound for the coefficient variations of $\mathrm{dd}^{\mathrm{c}}u_ j$. Bounded convergence gives $u_ j\to u$ in local $L^1$, and therefore $\mathrm{dd}^{\mathrm{c}}u_ j\to \mathrm{dd}^{\mathrm{c}}u$ in distributions. The uniform coefficient bounds show that this limit has measure coefficients with the asserted bound. $\square$