Pome6CBD4E4A94B6Proposition 5.28.5.6CBD4E4A94B6 Let $u_1,\ldots ,u_ k$ be locally $C^{1,1}$ quasi-plurisubharmonic functions, and let $\theta _1,\ldots ,\theta _ k$ be smooth real $(1,1)$-forms. Then the bounded product
is the form with locally bounded measurable coefficients obtained by taking the wedge product of the factors pointwise almost everywhere.
Proof. On a smaller coordinate ball, add a common smooth quadratic function $q$ such that each $u_ i+q$ is plurisubharmonic, and convolve with a smooth radial probability kernel. The Hessians of the convolutions are uniformly locally bounded and converge to the weak Hessians in every $L^ s$ space with $s<\infty $, so their finite products converge in $L^1$ on smaller balls. Since the potentials converge uniformly, the continuity of bounded Bedford–Taylor products identifies this limit with the pluripotential product. Expanding with respect to the smooth correction, as in (5.28.1), proves the assertion for the original factors. $\square$
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