Corollary 16.14.11.E3B684F226D1 Let $\theta _1,\ldots ,\theta _ n$ be closed smooth real $(1,1)$-forms on $X$, and let $f_1,\ldots ,f_ n$ be continuous real functions on $X$ with bounded metric Laplacian. Then each $\theta _ j+\mathrm{dd}^{\mathrm{c}}f_ j$ is a real $(1,1)$-form whose coefficients, computed from the weak second derivatives of $f_ j$, lie in $L^ p$ for every $p<\infty $. Their pointwise wedge product is defined almost everywhere and satisfies
\[ \bigwedge _{j=1}^ n(\theta _ j+\mathrm{dd}^{\mathrm{c}}f_ j) =F\, \mathrm dV,\qquad F\in L^ q(X)\quad \text{for every }1\leq q<\infty ; \]
in particular it is a finite signed Radon measure. On a coordinate ball, with smooth real $\rho _ j$ such that $\mathrm{dd}^{\mathrm{c}}\rho _ j=\theta _ j$, it equals $n!\pi ^{-n}D(B_1,\ldots ,B_ n)\, \mathrm dV_{2n}$, where $B_ j=((\rho _ j+f_ j)_{a\bar b})_{a,b}$. If moreover $u_ j$ are continuous $\theta _ j$-psh functions on $X$ with bounded metric Laplacian, then on $E=\bigcap _ j\{ u_ j=f_ j\} $ this signed measure agrees with the positive measure $\mathbf1_ E\bigwedge _ j(\theta _ j+\mathrm{dd}^{\mathrm{c}}u_ j)$ of proposition 16.14.6.
Proof.
By proposition 16.14.9 and (16.14.3), every $f_ j$ belongs to $W^{2,p}_{\mathrm{loc}}$ in every chart, for every finite $p$. In a chart, the current $\theta _ j+\mathrm{dd}^{\mathrm{c}}f_ j$ is therefore given by integration against the form $\theta _ j+\frac{\mathrm i}{2\pi }\sum _{a,b}(f_ j)_{a\bar b}\, \mathrm dz_ a\wedge \mathrm d\bar z_ b$, where the $(f_ j)_{a\bar b}$ are the weak derivatives, as in (5.2.9); its coefficients lie in $L^ p_{\mathrm{loc}}$. On the overlap of two charts, the forms obtained in the two charts represent the same current. For a current with locally integrable coefficients, (5.3.6), in which $\delta $ is the Jacobian determinant relating the two Lebesgue measures, says that the coefficients in the two charts are related, as distributions and hence almost everywhere (proposition 2.5.3(4)), by the pointwise change of variables of forms. Thus the two forms agree almost everywhere as pointwise forms. Hence the pointwise wedge product of the forms $\theta _ j+\mathrm{dd}^{\mathrm{c}}f_ j$ is a well-defined $(n,n)$-form, almost everywhere on $X$, and we write it as $F\, \mathrm dV$.
A coefficient of the wedge product in a chart is a sum of products of $n$ coefficients of the factors and of smooth functions; by Hölder's inequality, with each factor in $L^{nq}$ on compact subsets of the chart, these products lie in $L^ q$ there, for every $1\leq q<\infty $. Covering $X$ by finitely many compact subsets of charts, we obtain $F\in L^ q(X)$. A real function in $L^1(X)$ defines a finite signed Radon measure through its positive and negative parts.
On a coordinate ball, $\theta _ j+\mathrm{dd}^{\mathrm{c}}f_ j=\mathrm{dd}^{\mathrm{c}}(\rho _ j+f_ j)=\frac{\mathrm i}{2\pi }\sum _{a,b}(B_ j)_{ab}\, \mathrm dz_ a\wedge \mathrm d\bar z_ b$ almost everywhere, and the identity $\omega _{A_1}\wedge \cdots \wedge \omega _{A_ n}=D(A_1,\ldots ,A_ n)\, \omega _ I^ n$ of lemma 5.3.3(2), applied pointwise with $A_ j=B_ j/(2\pi )$, together with $\omega _ I^ n=n!\, 2^ n\, \mathrm dV_{2n}$, gives $\bigwedge _ j(\theta _ j+\mathrm{dd}^{\mathrm{c}}f_ j)=n!\pi ^{-n}D(B_1,\ldots ,B_ n)\, \mathrm dV_{2n}$. Finally, on every coordinate ball $W$ the second identity of proposition 16.14.6, applied with $U=X$, shows that $\mathbf1_ E\bigwedge _ j(\theta _ j+\mathrm{dd}^{\mathrm{c}}u_ j)=\mathbf1_ EF\, \mathrm dV$ on $W$. Since coordinate balls cover $X$, the two measures agree on $X$.
$\square$
Pome
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