Pome348FF027ECE6Lemma 8.11.10.348FF027ECE6 Every bounded admissible potential on $X$ is locally quasicontinuous for each smooth capacity $c_ s^*$, $0\leq s\leq m$. For $0\leq p<m$, each coefficient measure $\nu $ of the ordered $D_ p$ satisfies
uniformly when its global potentials have one common bound. The assertion at $p=0$ means the volume coefficient of $D_0=1$.
Proof. The global decreasing approximation and 8.9.8 provide bounded smooth local approximations to $u+\rho $. The preceding proposition makes its canonical pointwise limit quasicontinuous. Subtracting the smooth $\rho $ preserves quasicontinuity.
For smooth approximations write $\mathrm{dd}^{\mathrm{c}}u_ i=\mathrm{dd}^{\mathrm{c}}(u_ i+\rho )-\mathrm{dd}^{\mathrm{c}}\rho $ and expand the product. Each term has $q\leq p<m$ local cone factors and fixed smooth complementary factors. The absolute-form estimate and (8.11.3), with $s=m-1$, bound each coefficient variation by $C c_{m-1}$ on Borel sets. The constants depend only on the common bound and local data.
Take the iterated limits of 8.9.3 on a slightly larger compact neighborhood of $K$. Lower semicontinuity of signed measure variation on open sets preserves these estimates at each limit. For any open $G\supset E$, intersect with a fixed open neighborhood of $K$ having compact closure in $U$; this gives $|\nu |(E)\leq Cc_{m-1}(G)$. Taking the infimum proves the outer estimate. No simultaneous rough product or rough diagonal polynomial expansion is used. $\square$
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