Proposition 5.46.309F56C9DE42C

Proof. The two local inequalities show that the pole sets agree. Off $S$ put $w=u-v$, and give $w$ arbitrary bounded values on $S$. Since $S$ has Lebesgue measure zero, this defines the locally bounded distribution $u-v$. The coefficient measures of $\mathrm{dd}^{\mathrm{c}}w$ are locally finite, since after a smooth correction the complex Hessians of $u$ and $v$ are positive currents of order zero. Taking the Euclidean trace shows that $\Delta w$ is a signed Radon measure, so lemma 5.46.1 gives $w\in W^{1,2}_{\mathrm{loc}}$.

Consequently, the real and imaginary parts of the coefficient measures of $\mathrm{dd}^{\mathrm{c}}w$ are continuous linear functionals on $W^{1,2}_0$ of a smaller ball, since a second distributional derivative of $w$ is a first derivative of its $L^2$ gradient. Let $\nu $ be any one of these real signed measures, with Jordan decomposition $\nu =\nu ^+-\nu ^-$. Choose a compact set $K$ contained in the intersection of $S$ with a positive set of a Hahn decomposition, so that $\nu ^-(K)=0$. For $\epsilon >0$, take an open neighborhood $O$ of $K$ in the ball with $\nu ^-(O)<\epsilon $. The test functions $\varphi _ j$ of lemma 5.46.2, applied to $u$, satisfy

By continuity on the Sobolev space, the first term tends to zero. Letting $\epsilon \downarrow 0$ and using inner regularity gives $\nu ^+(S)=0$ locally, and the same argument applied to $-\nu $ gives $\nu ^-(S)=0$. Thus all coefficient measures of $\mathrm{dd}^{\mathrm{c}}w$ vanish on $S$, which proves the first identity. Since smooth forms also give $S$ zero mass, the second identity follows. $\square$