Theorem 15.25.1.8CE230CD94D4 Let $h$ be a Hermitian form on a connected compact complex manifold $X$. Suppose $u\in \mathrm{PSH}(X,h)\cap C^{0,\alpha }(X)$ for some $0<\alpha \leq 1$, and let $\nu $ be a finite nonzero positive measure with
\[ \nu \leq A\mathrm{MA}_ h(u),\qquad A>0. \]
There are a constant $c>0$ and a Hölder continuous $w\in \mathrm{PSH}(X,h)$ such that
\[ \sup _ Xw=0,\qquad \mathrm{MA}_ h(w)=c\nu . \]
In particular, its mass is finite and positive. No mass bound over all bounded $h$-plurisubharmonic functions is assumed.
Proof.
Step 1: approximate the density and reference. Write $\sigma =\mathrm{MA}_ h(u)$ and $\nu =A g\sigma $, with $0\leq g\leq 1$. Choose continuous $g_ l\in [0,1]$ tending to $g$ in $L^1(\sigma )$. Regularity of the finite measure supplies this approximation. The inequality $|s^{1/n}-t^{1/n}|^ n\leq |s-t|$ on $[0,1]$, followed by Hölder's inequality, gives $g_ l^{1/n}\to g^{1/n}$ in $L^1(\sigma )$.
By lemma 5.42.7, there are smooth $v_ j\in \mathrm{PSH}(X,h)$ tending uniformly to $u$ with a common Hölder norm. Their bounded products $\sigma _ j=h_{v_ j}^ n$ tend weakly to $\sigma $. Choose a dense countable family of continuous tests, including the constant one. For each $l$, choose $j(l)\geq l$ so that integration of $g_ l^{1/n}$, and of $g_ l$ multiplied by the first $l$ tests, against $\sigma _{j(l)}-\sigma $ has absolute value at most $1/l$. Write $\widehat v_ l=v_{j(l)}$ and
\[ \nu _ l=A g_ l h_{\widehat v_ l}^ n. \]
The common bounded-reference mass and $0\leq g_ l\leq 1$ give uniformly bounded masses. The diagonal and density approximation imply
\[ \nu _ l\rightharpoonup \nu ,\qquad \lim _ l\int _ Xg_ l^{1/n}h_{\widehat v_ l}^ n =\int _ Xg^{1/n}\, \mathrm d\sigma >0. \]
Discard finitely many terms if needed. Each $\nu _ l$ has a continuous density with respect to smooth volume and positive mass. Thus theorem 14.19.2 gives continuous normalized $h$-plurisubharmonic $\varphi _ l$ and $c_ l>0$ with $h_{\varphi _ l}^ n=c_ l\nu _ l$.
Step 2: control the coefficients and potentials. Since both right-hand sides have smooth-volume densities, lemma 5.28.7 gives
\[ h_{\varphi _ l}\wedge h_{\widehat v_ l}^{n-1} \geq (c_ lA)^{1/n}g_ l^{1/n}h_{\widehat v_ l}^ n. \]
The trace bound in lemma 5.29.20 is uniform for references with a fixed supremum norm, as its proof shows. Integrating therefore bounds $c_ l$ above. By proposition 5.29.13, the solution measures $\mu _ l=h_{\varphi _ l}^ n\leq C h_{\widehat v_ l}^ n$ have a common square-capacity bound.
Fix $\varepsilon =1/2$, the zero competitor, and $D>0$ with $2D<\delta _{1/2}$. The backwards estimate of lemma 14.19.16 gives
\[ \operatorname {Cap}_ h\{ \varphi _ l<\inf _ X\varphi _ l+D\} \geq a_0>0. \]
The normalized tail estimate (5.29.4) then bounds $-\inf _ X\varphi _ l$ uniformly. Applying lemma 14.19.14 at $s=t=D$ also gives
\[ D^ na_0\leq C\mu _ l(X)=Cc_ l\nu _ l(X). \]
Since the masses of $\nu _ l$ are uniformly bounded above, $c_ l$ is bounded away from zero.
Step 3: take a uniform limit. Extract a subsequence with $c_ l\to c>0$ and $\varphi _ l$ converging in $L^1(\mathrm dV)$, using theorem 5.25.2. All potentials now have common upper and lower bounds. The common Hölder norms of $\widehat v_ l$ and corollary 5.29.9 give, uniformly in $j,k$,
\[ \int _ X|\varphi _ j-\varphi _ k|\, \mathrm d\mu _ j \leq C\| \varphi _ j-\varphi _ k\| _{L^1(\mathrm dV)}^{\alpha /(\alpha +2n)} \longrightarrow 0. \]
Use proposition 14.19.23 in both orders to see that this subsequence is uniformly Cauchy. Its continuous limit $w$ is $h$-plurisubharmonic and has supremum zero. Uniform bounded-product continuity and $\nu _ l\rightharpoonup \nu $ yield $\mathrm{MA}_ h(w)=c\nu $. Finally theorem 15.5.4, with reference $u$ and domination constant $cA$, gives the Hölder assertion. Its mass is $c\nu (X)$.
$\square$
Pome
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