If the datum has finite moments of sufficiently high order against normalized $\theta $-plurisubharmonic functions, every normalized $u\in \mathcal E_\varphi $ whose measure is dominated by the datum differs from $\varphi $ by a bounded function. We work directly with $u\in \mathcal E_\varphi $: truncating $u$ against $\varphi -j$ would introduce a contact measure that need not be controlled by the datum. Envelopes with concave obstacles, as in lemma 10.31.14, keep this domination and give a differential inequality for the distribution of $\varphi -u$.
Lemma 10.31.53.F816A9EF0F9D Let $u\in \mathcal E_\varphi $ with $u\leq \varphi $, and let $\chi :\mathbb R_{\leq 0}\to \mathbb R_{\leq 0}$ be increasing and concave, with $\chi (0)=0$. Suppose that $\chi $ is $C^{1,1}$ on compact intervals and that, for a finite constant $L\geq 1$,
\[ 1\leq \chi '\leq L. \]
Then the envelope
\[ p=P_\theta (\varphi +\chi (u-\varphi )) \]
belongs to $\mathcal E_\varphi $, and
\[ \mu _ p\leq \mathbf1_{\{ p=\varphi +\chi (u-\varphi )\} } \chi '(u-\varphi )^ n\mu _ u. \]
Here the obstacle and the contact set are defined off the pluripolar set where $u-\varphi $ is undefined, which all these measures ignore.
Proof.
For $t\leq 0$, the bounds on $\chi '$ give $Lt\leq \chi (t)\leq t$. Thus
\[ Q_ L(u,\varphi )\leq p\leq u\leq \varphi . \]
The lower potential belongs to $\mathcal E_\varphi $ by lemma 16.28.63, so $p$ is not identically $-\infty $ and belongs to $\mathcal E_\varphi $ by upward stability. The obstacle is finite and quasi-continuous off the poles, so the contact theorem theorem 15.8.6 applies.
Let $\tau =\chi ^{-1}$. It is convex, with $L^{-1}\leq \tau '\leq 1$. We justify the transformation
\[ w=\varphi +\tau (p-\varphi ) \]
without assuming that $\varphi $ is locally bounded on some open set. On a coordinate ball choose a smooth function $q$ with $\theta +\mathrm{dd}^{\mathrm{c}}q\geq 0$. The functions
\[ p_ m=\max (p,q-m),\qquad \varphi _ m=\max (\varphi ,q-m) \]
are locally bounded $\theta $-plurisubharmonic functions on the ball, and $p_ m\leq \varphi _ m$. Lemma 5.28.13 applies to $w_ m=\varphi _ m+\tau (p_ m-\varphi _ m)$ and gives
\[ \theta _{w_ m}\geq 0,\qquad \mu _{w_ m}\geq \tau '(p_ m-\varphi _ m)^ n\mu _{p_ m}. \]
The function $(s,t)\mapsto t+\tau (s-t)$ on $\{ s\leq t\} $ is nondecreasing in each variable. Consequently $w_ m$ decreases as $m$ increases. Since $p\leq p_ m\leq w_ m\leq \varphi _ m$, the limit is not identically $-\infty $ on the ball. Off the poles it equals $w$. The local limits therefore agree on overlaps after regularization and define a global $\theta $-plurisubharmonic function.
On $D_ m=\{ p>q-m\} $, neither $p$ nor $\varphi $ is truncated. Thus $w_ m=w$ there, and plurifine locality transfers the product inequality to $D_ m$. These sets exhaust the complement of the poles of $p$ in the ball. Since all products ignore the discarded pluripolar set,
\[ \mu _ w\geq \tau '(p-\varphi )^ n\mu _ p. \]
The obstacle bound $p\leq \varphi +\chi (u-\varphi )$ gives $w\leq u$, with equality on the contact set of $p$. Ordered contact, the identity $\tau '(p-\varphi )=1/\chi '(u-\varphi )$ on that set, and contact support of $\mu _ p$ prove the assertion.
$\square$
Proposition 10.31.54.36BCEF7C4255 Let $\mu $ be a finite non-pluripolar measure. Suppose that, for some real $m>2n+1$,
\[ A_ m:=\sup _{\substack {v\in \mathrm{PSH}(X,\theta )\\ \sup _ Xv=0}} \int _ X(-v)^ m\, \mathrm{d}\mu <\infty . \]
There is a finite constant $C$, depending only on $n,\ell ,M,m,\mu (X)$ and $A_ m$, such that every $u\in \mathcal E_\varphi $ with $\sup _ Xu=0$ and $\mu _ u\leq \mu $ satisfies
\[ \varphi -C\leq u\leq \varphi . \]
Proof.
As in proposition 10.31.15, we build a concave weight from the distribution of $\varphi -u$ and apply the preceding lemma. The moment bound on $\mu $ now yields a differential inequality for this weight, which bounds the depth of $\varphi -u$ under $\mu _ u$.
The envelope identity gives $u\leq \varphi \leq 0$. Put $x=\varphi -u$, defined off the poles, and $N=2n+1$. Since $x\leq -u$ and $\mu _ u\leq \mu $, Hölder's inequality gives the uniform first-moment bound
\[ B:=\int _ Xx\, \mu _ u \leq \int _ X(-u)\, \mathrm{d}\mu \leq \mu (X)^{1-1/m}A_ m^{1/m}=:A_1. \]
Let $H(t)=\mu _ u(\{ x>t\} )$ and $T=\sup \{ t\geq 0:H(t)>0\} $, with $T=0$ if this set is empty; at this point $T$ may be infinite. If $T<\infty $, zero domination applied to $u$ and $\varphi -t$ for $t>T$ gives $u\geq \varphi -T$. It is therefore enough to bound $T$. We may assume $T>1$, and we choose any $S$ with $1<S<T$.
Define, for $t\geq 0$,
\[ g(t)= \begin{cases} ((1+t)^{3/2}H(t))^{-1},& 0\leq t\leq S,\\ (1+t)^{-3/2},& t>S, \end{cases} \qquad F(t)=1+\int _0^ t g(s)\, \mathrm{d}s, \]
and set $W(t)=\int _0^ t F(s)^{1/N}\, \mathrm{d}s$ and $\chi (-t)=-W(t)$. Since $H(S)>0$, $F$ is bounded on the whole half-line, although its bound may depend on $S$ and $u$. It is locally Lipschitz and nondecreasing. Hence $\chi $ has bounded derivative, is increasing and concave, and satisfies the hypotheses of the preceding lemma. Also $W(t)\geq t$ and $W(t)\leq tF(t)^{1/N}$.
Since $H(t)\leq M$, Tonelli's theorem gives a bound independent of $S$ and $u$:
\[ \int _ X F(x)\, \mu _ u =\mu _ u(X)+\int _0^\infty g(t)H(t)\, \mathrm{d}t \leq 3\max (1,M)=:K. \]
Let $p=P_\theta (\varphi -W(x))\in \mathcal E_\varphi $, the envelope of the preceding lemma for this $\chi $. On its contact set the lemma bounds its measure by $F(x)^{n/N}\mu _ u$. Hence, using $N=2n+1$ and the Cauchy–Schwarz inequality,
\[ \begin{split} \int _ X(\varphi -p)^{1/2}\, \mu _ p & \leq \int _ X x^{1/2}F(x)^{(n+1/2)/N}\, \mu _ u\\ & \leq \left(\int _ Xx\, \mu _ u\right)^{1/2} \left(\int _ XF(x)\, \mu _ u\right)^{1/2} \leq \sqrt{KB}. \end{split} \]
The identity $P_\theta [p]=\varphi $ implies $p-\sup _ Xp\leq \varphi $. Since $p\leq \varphi \leq 0$ and $\mu _ p(X)\geq \ell $, the last integral is at least $\ell \sqrt{-\sup _ Xp}$. Consequently
\[ 0\leq -\sup _ Xp\leq KA_1/\ell ^2=:D. \]
Using $W(x)\leq \varphi -p\leq -p$, and applying the moment hypothesis to $p-\sup _ Xp$, we obtain
9E93E3C5CAED
10.31.55
\begin{equation} \int _ XW(x)^ m\, \mathrm{d}\mu \leq 2^{m-1}\bigl (A_ m+D^ m\mu (X)\bigr )<C_0, \end{equation}
where we choose $C_0=1+2^{m-1}(A_ m+D^ m\mu (X))$. This constant is independent of $S$ and $u$.
For $0<t\leq S$, Chebyshev's inequality and $\mu _ u\leq \mu $ give
\[ H(t)\leq C_0W(t)^{-m}, \qquad F'(t)\geq \frac{W(t)^ m}{C_0(1+t)^{3/2}} \quad \text{for almost every }t. \]
Since $W'=F^{1/N}$, the derivative of $(W')^{N+1}$ is at least
\[ \frac{N+1}{NC_0}\, \frac{W^ mW'}{(1+t)^{3/2}}. \]
All functions being differentiated are absolutely continuous on compact intervals. Integrating from $0$ to $t$ and using $W(0)=0$ gives
\[ (W'(t))^{N+1} \geq \frac{N+1}{NC_0(m+1)} \frac{W(t)^{m+1}}{(1+t)^{3/2}}. \]
In particular
\[ W'(t)\geq c_0(1+t)^{-\gamma }W(t)^\kappa , \quad \gamma =\frac{3}{2(N+1)}<1,\quad \kappa =\frac{m+1}{N+1}>1, \]
with $c_0>0$ independent of $S$ and $u$. Integrating the derivative of $W^{1-\kappa }$ over $[1,S]$ yields
\[ (\kappa -1)c_0\int _1^ S(1+t)^{-\gamma }\, \mathrm{d}t \leq W(1)^{1-\kappa }-W(S)^{1-\kappa }\leq 1, \]
because $W(1)\geq 1$. Since the integral on the left diverges as $S\to \infty $, every $S<T$ is bounded by a constant independent of $S$ and $u$, and hence so is $T$. The zero-domination argument at the beginning of the proof now gives $u\geq \varphi -C$. The case $T\leq 1$, which includes $B=0$, was covered there as well.
$\square$
Corollary 10.31.56.B2E2FCD34986 Let $0\leq f\in L^ p(X,\mathrm{d}V)$, with $p>1$ and $\int _ Xf\, \mathrm{d}V>0$. For $\lambda >0$, the solution in theorem 10.31.44 of
\[ \mathrm{MA}_\theta (u)=\mathrm{e}^{\lambda u}f\, \mathrm{d}V \]
satisfies $u\sim \varphi $. Every normalized solution in $\mathcal E_\varphi $ of
\[ \mathrm{MA}_\theta (u)=cf\, \mathrm{d}V,\qquad c>0, \]
also satisfies $u\sim \varphi $. In particular, both existence statements hold in the ordinary singularity type of $\varphi $.
Proof.
Lemma 5.26.2, applied in a Hermitian background dominating $\theta $, gives uniform moments of every finite order against $\mathrm{d}V$ for normalized $\theta $-plurisubharmonic functions. Hölder's inequality therefore implies, for every finite $m$,
\[ \sup _{\substack {v\in \mathrm{PSH}(X,\theta )\\ \sup _ Xv=0}} \int _ X(-v)^ m f\, \mathrm{d}V<\infty . \]
Choose $m>2n+1$. For the solution of the exponential equation, set $S=\sup _ Xu$ and $v=u-S$. Then $v\in \mathcal E_\varphi $, $\sup _ Xv=0$, and
\[ \mu _ v=\mathrm{e}^{\lambda u}f\, \mathrm{d}V \leq \mathrm{e}^{\lambda S}f\, \mathrm{d}V. \]
The preceding proposition, applied to this fixed dominating measure, shows that $v$, and hence $u$, has the ordinary type of $\varphi $. For the normalized equation, apply the proposition to the measure $cf\, \mathrm{d}V$.
$\square$
Pome
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