Theorem 5.27.42.745A94C03209 Assume that $\Omega $ is bounded, $n\geq 2$, and $b\in \mathrm{PSH}(\Omega )\cap L^\infty (\Omega )$. For a set $F\subset \mathcal M_+(\Omega )$, the following are equivalent:
$F$ is vaguely closed and every $\mu \in F$ satisfies $\mu \geq (\mathrm{dd}^{\mathrm{c}}b)^ n$;
there are bounded plurisubharmonic functions $u_ j\leq b$ on $\Omega $, converging to $b$ in $L^ p(\Omega )$ for every $1\leq p<\infty $ and in local capacity, whose full vague cluster set is $F$.
The empty set is allowed. If $b$ is smooth, every $u_ j$ in the second condition can be chosen smooth.
Proof.
If the second condition holds, lemma 5.27.41 gives the measure inequality, and the full cluster set is closed in the metrizable vague topology by the argument at the beginning of the proof of theorem 5.27.39.
Conversely, assume the first condition. If $F$ is nonempty, the concatenation of a dense sequence used in the proof of theorem 5.27.39 gives measures $\nu _ j\in F$ whose full vague cluster set is $F$. If $F$ is empty, take $\nu _ j=(\mathrm{dd}^{\mathrm{c}}b)^ n+j\delta _ a$ for a fixed $a\in \Omega $. A nonnegative compactly supported test function equal to one at $a$ has integral at least $j$ against $\nu _ j$, so this sequence has no vaguely convergent subsequence. In either case, $\nu _ j-(\mathrm{dd}^{\mathrm{c}}b)^ n$ is a positive Radon measure.
Let $(f_ i)$ and $d_ v$ be as in lemma 5.27.38. For each $j$, approximate the excess $\nu _ j-(\mathrm{dd}^{\mathrm{c}}b)^ n$ against $f_1,\ldots ,f_ j$ by a positive finite atomic measure, with each error less than $1/j$. If this measure has $N_ j>0$ atoms, perform the construction from the proof of theorem 5.27.36, starting at $b$ and using $\tau =t=(jN_ j)^{-1}$ for each atom; if it has none, take $u_ j=b$. Summing the estimates of proposition 5.27.35 gives
\[ \| u_ j-b\| _{L^ p(\Omega )}\leq C_{\Omega ,p}/j, \qquad \left|\int _\Omega f_ i\bigl ((\mathrm{dd}^{\mathrm{c}}u_ j)^ n-\nu _ j\bigr )\right| <2/j\quad (i\leq j). \]
For every fixed pair of concentric balls $B'\Subset B\Subset \Omega $, the same construction bounds the integral of $|u_ j-b|$ over $B'$ against $(\mathrm{dd}^{\mathrm{c}}w)^ n$ by $C_{B',B,\Omega }/j$, for every $w\in \mathrm{PSH}(B)$ with $0\leq w\leq 1$. Dividing by a fixed positive threshold and taking the supremum over $w$ gives convergence in local capacity. Each $u_ j$ is bounded and lies below $b$, and it is smooth when $b$ is smooth. One sequence serves all finite $p$, and no common local mass bound on the $\nu _ j$ is needed.
The error in the first $j$ coordinates and the tail of the series yield
\[ d_ v\bigl ((\mathrm{dd}^{\mathrm{c}}u_ j)^ n,\nu _ j\bigr )\leq 2/j+2^{-j}\longrightarrow 0. \]
By the remark following lemma 5.27.38, the two sequences of measures have the same limits along corresponding subsequences. Hence their full vague cluster sets coincide.
$\square$
Pome
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.