PomeC5B4B944142CLemma 5.28.10.C5B4B944142C Let $\vartheta $ be a smooth real $(1,1)$-form on $U$. Suppose that $u$ is continuous, $T=\vartheta +\mathrm{dd}^{\mathrm{c}}u\geq 0$, and $T^ n\geq g^ n\beta ^ n$ with $g\geq 0$ continuous. For a nonnegative convolution kernel $\chi _\delta $ of mass one,
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5.28.11
on every smaller domain on which the convolution is defined.
Proof. Approximate the convolution by finite convex combinations of translates, and apply (5.28.9), extended by induction to finitely many terms. Along these approximations, the background forms converge smoothly and the potentials converge uniformly on compact sets. To pass to the limit, add a common quadratic function and write the difference of two $n$-th powers as a telescoping sum. After pairing with a test form, integration by parts bounds each term by the uniform distance between the potentials times the mass of a mixed current, and successive integration against nested cutoff functions bounds these masses uniformly. Thus the products of the bounded potentials converge, and (5.28.11) follows. $\square$
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