Pome37976A2408E0Convolution in coordinate charts gives smooth approximations of a finite measure. For a varying quasi-plurisubharmonic integrand, the submean inequality transfers the error to the original measure; this gives the additional control needed beyond weak convergence of the measures themselves.
For a smooth real $(1,1)$-form $\theta $ with $\mathrm{PSH}(X,\theta )\ne \varnothing $ and a non-pluripolar probability measure $\nu $ on $X$, put
This number may be infinite. It is finite for Monge–Ampère measures of bounded potentials.
Lemma 5.44.10.2E102F06172B Let $\psi \in \mathrm{PSH}(X,\omega )$ be bounded, and put $\nu =(\omega +\mathrm{dd}^{\mathrm{c}}\psi )^ n$. There is a finite constant $C$, depending only on $\omega $, $\theta $, $\mathrm{d}V$ and $\| \psi \| _\infty $, such that
Proof. Choose finitely many pairs of concentric coordinate balls $B'_ a\Subset B_ a$ such that the $B'_ a$ cover $X$, and fixed smooth functions $q_ a,r_ a$ on $B_ a$ with $\mathrm{dd}^{\mathrm{c}}q_ a\geq \omega $ and $\mathrm{dd}^{\mathrm{c}}r_ a\geq \theta $, with $r_ a\leq 0$. Then $\psi +q_ a$ is bounded and plurisubharmonic, $w+r_ a$ is negative and plurisubharmonic, and positivity of bounded products gives
The last step is the weighted Chern–Levine–Nirenberg estimate of section 5.27. Summing over the cover and using the normalized $L^1$ bound of theorem 5.25.2 proves the lemma. $\square$
To construct the smoothings, choose finitely many coordinate balls $B_ a$ and smooth functions $\eta _ a\geq 0$ with compact support in $B_ a$ and $\sum _ a\eta _ a=1$. Let $\chi \geq 0$ be a smooth radial kernel of integral one supported in the unit ball of $\mathbb C^ n$, and fix $\epsilon _ j\downarrow 0$ so small that all convolutions below stay in the corresponding coordinate ball. For a function $F$, let $S_{a,j}F$ be its convolution, in the chart of $B_ a$, with the kernel $\epsilon _ j^{-2n}\chi (\cdot /\epsilon _ j)$. For a probability measure $\nu $ on $X$, define a smooth probability measure $\nu _ j$ by convolving each coordinate measure $\eta _ a\nu $ with the same kernel and adding the extensions by zero. It is characterized by the identity
D656A5C5B2DC
5.44.11
for Borel functions $F$; since the kernel is even, the same convolution occurs on the right. This identity first holds for nonnegative Borel functions and then for integrable signed functions. In particular, $\nu _ j\rightharpoonup \nu $ weakly.
Lemma 5.44.12.88E063207C1B Let $\nu $ be a non-pluripolar probability measure with $A_\theta (\nu )<\infty $, and let $\nu _ j$ be its smoothings defined by (5.44.11). Then
Moreover, let $z_ j,z$ be quasi-plurisubharmonic functions with a common upper bound and a common smooth lower curvature bound, and suppose that $z_ j\to z$ both in $L^1(X)$ and in $L^1(\nu )$. Then
2D0225A9A28D
5.44.13
Consequently, for every $\lambda >0$,
Proof. On each coordinate ball $B_ a$ choose a fixed smooth function $q_ a$ with $\mathrm{dd}^{\mathrm{c}}q_ a\geq \theta $. If $w\in \mathrm{PSH}(X,\theta )$ and $\sup _ Xw=0$, then $w+q_ a$ is plurisubharmonic, and the submean inequality gives
with a constant independent of $w$ and $j$. Integrating and using (5.44.11) proves $A_\theta (\nu _ j)\leq A_\theta (\nu )+C$.
For the second assertion, the tail envelopes
decrease to $z$ by lemma 5.25.1. Enlarge the functions $q_ a$ so that $H_ j+q_ a$, $z_ j+q_ a$ and $z+q_ a$ are all plurisubharmonic. Since $H_ j\geq z_ j$ and $H_ j\geq z$, we have $|z_ j-z|\leq 2H_ j-z_ j-z$, and the identity (5.44.11) and the submean inequality imply
All these integrals are well defined: each function on the larger side of a submean inequality is bounded above, and its negative part is controlled by that of the integrable function on the smaller side plus a constant.
Radial averages of a plurisubharmonic function decrease when the radius decreases. Therefore $S_{a,j}(H_ j+q_ a)$ decreases to $z+q_ a$ on the support of $\eta _ a$: to identify the limit, fix an index $k$, use $H_ j\leq H_ k$ for $j\geq k$, let $j\to \infty $ in the radius, and then let $k\to \infty $; the reverse inequality is the submean inequality. The nonnegative integrand in the second line of the last display is dominated on this compact support by a constant minus $z$, which is $\nu $-integrable, so its integral tends to zero by dominated convergence. The final integral tends to zero by convergence in $L^1(\nu )$. This proves (5.44.13).
Since the exponential is Lipschitz on the common range of the functions, (5.44.13) also gives $\int _ X|\mathrm{e}^{\lambda z_ j}-\mathrm{e}^{\lambda z}|\, \mathrm{d}\nu _ j\to 0$. If $\zeta >0$ is smooth, then $\zeta \mathrm{e}^{\lambda z}=\mathrm{e}^{\lambda z+\log \zeta }$ is bounded and quasi-plurisubharmonic. For any fixed bounded quasi-plurisubharmonic function $F$, the same local submean argument and dominated convergence in (5.44.11) give
Applying this to $F=\zeta \mathrm{e}^{\lambda z}$ gives $\int _ X\zeta \mathrm{e}^{\lambda z_ j}\, \mathrm{d}\nu _ j\to \int _ X\zeta \mathrm{e}^{\lambda z}\, \mathrm{d}\nu $ for every smooth $\zeta >0$. Since every smooth function is a difference of two positive smooth functions, the convergence holds for all smooth test functions, and uniformly bounded masses together with uniform approximation of continuous functions by smooth ones prove weak convergence. $\square$
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