5.44.6 Measures with bounded potentials80117515F812

Proof. By lemma 5.25.1, the tail envelopes $A_ j=(\sup _{\ell \geq j}a_\ell )^*$ decrease to $a$. Since $a\leq \max (a_ j,a)\leq A_ j$, monotone convergence in capacity of bounded potentials (lemma 5.29.19) implies

We split $|a_ j-a|=(\alpha _ j-a)+(\alpha _ j-a_ j)$. The integral of the first part against the moving products tends to zero by lemma 5.29.16.

For the second part, put $h_ j=\alpha _ j-a_ j\geq 0$. We compare its integrals against the measures with indices $j$ and $\ell $. The case $k=0$ follows from the assumed $L^1$ convergence. For $k\geq 1$, put $z=w_ j-w_\ell $ and

Factorization and Stokes' theorem on the compact manifold $X$ give

Since $\omega $ is not assumed to be closed, $T$ need not be closed, and the product rule reads

None of the terms involving derivatives of $T$ need vanish.

We estimate the four terms. Since $\mathrm{dd}^{\mathrm{c}}h_ j=\omega _{\alpha _ j}-\omega _{a_ j}$, the absolute contribution of the first is at most

It tends to zero as $j,\ell \to \infty $ by lemma 5.29.16, since $w_ j-w_\ell \to 0$ in capacity. Differentiating each factor shows that $\mathrm{d}T=\mathrm{d}\omega \wedge R$ and $\mathrm d^{\mathrm c}T=\mathrm d^{\mathrm c}\omega \wedge R$, where $R$ is a positive linear combination of mixed products of bidegree $(n-2,n-2)$. Moreover, $\mathrm{dd}^{\mathrm{c}}T$ is a sum of terms consisting of $\mathrm{dd}^{\mathrm{c}}\omega $ times mixed products of degree $n-2$, or of $\mathrm{d}\omega \wedge \mathrm d^{\mathrm c}\omega $ times mixed products of degree $n-3$; terms of negative degree are absent. These smooth torsion factors are bounded in absolute value by fixed multiples of $\omega ^2$ and $\omega ^3$, respectively. As $h_ j$ is uniformly bounded, the contribution of $h_ j\mathrm{dd}^{\mathrm{c}}T$ also tends to zero, by the same capacity estimate.

For the two gradient terms, the pointwise Cauchy–Schwarz inequality for a positive mixed product gives, for example,

The second factor tends to zero, and the first is uniformly bounded. Indeed, if $p$ is any of the uniformly bounded potentials and $S$ any positive mixed product of degree $n-1$ occurring here, then

Expand $\mathrm{dd}^{\mathrm{c}}S$ with all its torsion terms as above, and write $\mathrm{dd}^{\mathrm{c}}p=\omega _ p-\omega $. Boundedness of $p$ and lemma 5.29.16 bound both integrals uniformly. Finally, $\mathrm{d}h_ j\wedge \mathrm d^{\mathrm c}h_ j\leq 2\mathrm{d}\alpha _ j\wedge \mathrm d^{\mathrm c}\alpha _ j+2\mathrm{d}a_ j\wedge \mathrm d^{\mathrm c}a_ j$ as positive quadratic forms, which proves the asserted bound. The other gradient term is estimated in the same way. In dimension one, $T=1$ and all torsion terms in (5.44.8) vanish.

These integration-by-parts and energy identities hold for smooth potentials. They extend to bounded potentials through smooth decreasing approximations in the Hermitian background, using the local Bedford–Taylor chain rule, its gradient products and quasi-continuity; equivalently, they follow from the bounded current calculus of section 5.28. Only the common bound on the potentials enters the estimates.

We have shown that the difference between the integrals of $h_ j$ against the measures with indices $j$ and $\ell $ tends to zero as $j,\ell \to \infty $. For a fixed index $\ell $, the measure $\omega _{w_\ell }^ k\wedge \omega ^{n-k}$ is a fixed finite non-pluripolar measure, and lemma 5.44.2, applied separately to $\alpha _ j\to a$ and $a_ j\to a$ in $L^1(X)$, gives

Choosing first $\ell $ large and then letting $j\to \infty $, we conclude that the integrals of $h_ j$ against $\omega _{w_ j}^ k\wedge \omega ^{n-k}$ tend to zero. Together with the estimate for $\alpha _ j-a$, this proves the result. If the $a_ j$ only have some other fixed lower curvature bound, choose a Hermitian form $\Omega \geq \omega $ such that they are $\Omega $-plurisubharmonic. By section 5.29, convergence in capacity is unchanged by this replacement. Apply the result just proved with $\Omega $ in place of $\omega $; by positivity, the original mixed measures are bounded by their counterparts for $\Omega $. $\square$