Proposition 20.3.13.B3BE3C08418A The measures $D_ m$ satisfy $D_ m\leq D_ u$ and converge to $D_ u$ in total variation, and they increase along dyadic indices. For a fixed nonpositive Green kernel, the potentials $w_{2^ j}$ decrease to $w$, and $w_ m\to w$ in $L^1(Y)$. Each $w_ m$ has the ordinary singularity type of $u_ m|_ Y$.
If the restriction $u|_ Y$ is not identically $-\infty $, it is at least as singular as the atomic trace:
\[ u|_ Y\preceq \operatorname {Tr}^{\mathrm{at}}_ Y(u). \]
If $u$ has analytic singularities, the two are ordinarily equivalent.
Proof.
The supremum formula in (20.3.3) gives $D_ m\leq D_ u$ coefficient by coefficient. Since the coefficients converge and are dominated by those of the finite measure $D_ u$, dominated convergence gives $D_ m(Y)\to D_ u(Y)$, and hence convergence in total variation. Superadditivity with $s=t=m$ gives $D_{2m}\geq D_ m$, and since $G\leq 0$ the potentials $w_{2^ j}$ decrease. Monotone convergence for $-G$ identifies their pointwise limit with $w$, including at poles. The uniform $L^1$ bound for the kernel also gives
\[ \| w_ m-w\| _{L^1(\eta )} \leq C\| D_ m-D_ u\| _{\mathrm{TV}}\longrightarrow 0. \]
Each $D_ m$ is finitely supported, and locally both $w_ m$ and $u_ m|_ Y$ are the sum of the logarithmic singularities prescribed by $D_ m$ and a bounded function. Thus their ordinary types agree.
If $a=u|_ Y$ is not identically $-\infty $, proposition 20.3.6 gives $\nu (a,p)\geq \tau _{Y,u}(p)$ at every point. Since $\mathrm{dd}^{\mathrm{c}}w=D_ u-D_ u(Y)\eta $, lemma 5.38.5 implies $a\preceq w$. If $u$ has analytic singularities, proposition 20.3.6 identifies the finitely many logarithmic coefficients of $u|_ Y$ with those of $D_ u$, and the bounded local remainders give ordinary equivalence.
$\square$
Pome
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