5.27.32 Adding a prescribed local measure38F5E7BC8433

Proof. Write $S=\mathrm{dd}^{\mathrm{c}}v$, $q=\varepsilon \log (|\zeta |^2/R^2)$ and $T_ A=v-A-q$. The rank-one calculation in the proof of lemma 5.27.29 gives

off $D$; since $h_ A$ is smooth and $D$ has measure zero, this formula determines the current. Fix a smooth compactly supported test form. On the projection of its support to $U$, the function $v$ and its derivatives are bounded. Hence the support of $\chi _\eta '(T_ A)$ lies in $\{ |\zeta |\leq s_ A\} $, where $s_ A=C\exp (-A/(2\varepsilon ))$ and $C$ depends only on this compact set and the fixed data. The coefficients involving $v$ and $\chi _\eta $ are bounded at the fixed parameters, so the first summand has pairing $O(s_ A^2)$. In the second summand, the term with two derivatives of $v$ also has pairing $O(s_ A^2)$, and the two cross terms have pairing $O(s_ A)$: the derivative of $q$ in the $\zeta $ direction has size $O(|\zeta |^{-1})$, and its area integral over the disk $\{ |\zeta |\leq s_ A\} $ is $O(s_ A)$.

It remains to integrate the term containing $\mathrm dq\wedge \mathrm d^{\mathrm c}q$. The logarithmic change of the radial variable used for (5.27.30) leaves the factor $\varepsilon $ and the probability weight

Only the component of the test form in the $z'$ directions enters this term, and on the shrinking circles its coefficients tend uniformly to their values at $\zeta =0$. Integration in the angle therefore gives, in the limit, exactly the pairing with $\varepsilon [D]\wedge S^{r-1}$. This proves the convergence in every degree. $\square$

Proof. Put $M=1+\| b\| _{L^\infty (L)}$. Cover $K$ by finitely many balls whose larger concentric balls have closures in $L^\circ $. Replacing $b$ by $b+M$, which has the same Hessian and takes values in $[0,2M]$ on $L$, we may assume $0\leq b\leq 2M$ there. For nested cutoff functions $\psi $, integration by parts against closed forms gives

where $W$ is $T$ multiplied by the remaining positive closed factors. Apply this successively to the $n-r$ Hessians of $b$. For bounded $b$, the integrations by parts are justified by local decreasing regularization and theorem 5.27.1, with one common bound on the range. Summing over the finite cover gives the asserted constant. $\square$

Proof. We take for $h$ a function from lemma 5.27.29 whose top-degree measure approximates $c\delta _ a$ and whose lower-degree currents are small; lemma 5.27.34 then controls the mixed terms.

Write $a=(a_1,a')$ and $m=n-1$, and fix $R=2\sup _\Omega |z_1|+2$. For $\delta >0$, put $H_\delta (z')=\log (|z'-a'|^2+\delta ^2)$. By the radial calculation in the proof of proposition 5.27.31, $(\mathrm{dd}^{\mathrm{c}}H_\delta )^ m$ is a probability measure tending to $\delta _{a'}$ as $\delta \downarrow 0$. Extend every $f_ i$ by zero outside $\Omega $, and choose $\delta $ such that

Choose compact sets $K,L\subset \Omega $ with $K\subset L^\circ $ and $K$ containing the supports of all $f_ i$, and a nonnegative smooth cutoff function $\psi $ with compact support in $\Omega $, equal to one near $L$. Put $\beta =\mathrm{dd}^{\mathrm{c}}|z|^2$. Expanding the top power of $\mathrm{dd}^{\mathrm{c}}(b+h)$, summing the bounds of lemma 5.27.34 with their binomial coefficients and the norms $\| f_ i\| _\infty $, we find $\gamma >0$ such that the mixed terms contribute less than $\tau /3$ to every test integral whenever $h$ is smooth and plurisubharmonic and

Fix $0<\eta \leq t/2$. Apply lemma 5.27.29 on $\mathbb C\times \mathbb C^ m$ with transverse variable $\zeta =z_1-a_1$ and $v=c^{1/m}\varepsilon ^{-1/m}H_\delta $. For $r<n$, lemma 5.27.33 gives

as $A\to \infty $. For the fixed $\delta $, the integrals of these limits against $\psi \beta ^{n-r}$ are finite, and their exponents of $\varepsilon $ are positive. Choose $0<\varepsilon \leq t/2$ so that each of them is less than $\gamma /2$. With these parameters fixed, choose $A$ so large that the integrals $\int _ L(\mathrm{dd}^{\mathrm{c}}h_ A)^ r\wedge \beta ^{n-r}$ are less than $\gamma $, that $v-A\leq 0$ on the projection of $\Omega $ to the $z'$ variables, and that

The last choice follows directly from (5.27.30), because the circles there shrink uniformly over the projected supports of the tests and $\varepsilon (\mathrm{dd}^{\mathrm{c}}v)^ m=c(\mathrm{dd}^{\mathrm{c}}H_\delta )^ m$.

Set $h=h_ A-\eta $. As in the proof of proposition 5.27.31, $h$ is smooth, bounded and nonpositive on $\Omega $, with

The logarithm has $L^ p(\Omega )$ norm bounded uniformly for $a_1$ in the projection of $\Omega $ to the first coordinate: after translation, its one-variable integral is bounded by polar integration on one fixed disk. This uniform bound, the weighted Chern–Levine–Nirenberg estimate and the range estimate (5.27.3) give the asserted bound for the integrals over $B'$. Since $\varepsilon +\eta \leq t$, both estimates hold with constants depending only on $\Omega $, $p$, $B'$ and $B$.

Finally, multilinearity of bounded products gives

The choice of $\delta $, the choice of $A$ for the top-degree term and the estimate of the mixed terms each contribute less than $\tau /3$ to every test integral. This proves the required inequalities, with one choice of $h$ for all finite $p$. $\square$

Proof. Choose relatively compact open sets exhausting $\Omega $, each with closure contained in the next one. On each of these sets, take a countable family of smooth compactly supported functions that is uniformly dense in the space of continuous functions vanishing at its boundary, and add a nonnegative smooth compactly supported cutoff function equal to one on its closure. Enumerate the union of these families as $(f_ i)$. If positive measures converge against every $f_ i$ to a fixed positive Radon measure, they converge vaguely: the cutoff functions give common local mass bounds, and uniform approximation then handles every continuous compactly supported test function.

For each $j$, approximate $\nu $ against $f_1,\ldots ,f_ j$ by a finite positive atomic measure

Such an approximation uses only the finite mass of $\nu $ on a compact neighborhood of the supports of these tests: a finite partition on which all the tests have small oscillation provides the atoms and their positive weights. Atoms of weight zero are omitted. If $N_ j=0$, set $u_ j=b$. Otherwise start with $b_{j,0}=b$ and apply proposition 5.27.35 successively to $b_{j,s-1}$, $a_{j,s}$ and $c_{j,s}$, with the tests $f_1,\ldots ,f_ j$ and $\tau =t=(jN_ j)^{-1}$. This gives perturbations $h_{j,s}$, and we define

Every intermediate potential is bounded and plurisubharmonic, so the proposition applies at each step; the choice at the next step may depend on the bound of the current potential, and each perturbation serves all finite $p$ at once. Summing the norm estimates yields $\| u_ j-b\| _{L^ p(\Omega )}\leq C_{\Omega ,p}/j$, and since every $h_{j,s}\leq 0$, we have $u_ j\leq b$. Summing the measure errors gives

By the choice of the family $(f_ i)$, this proves the asserted vague limit, also when $\nu (\Omega )=\infty $.

For fixed concentric balls $B'\Subset B\Subset \Omega $, the perturbations are nonpositive, so $|u_ j-b|=\sum _ s|h_{j,s}|$. The last estimate of proposition 5.27.35 bounds the integral of this function against $(\mathrm{dd}^{\mathrm{c}}w)^ n$ over $B'$ by $C_{B',B,\Omega }/j$, for every $w\in \mathrm{PSH}(B)$ with $0\leq w\leq 1$. Restricting to the set where $|u_ j-b|$ exceeds a fixed threshold, dividing by the threshold and taking the supremum over $w$ proves convergence in local capacity. If $b$ is smooth, the finite sums defining $u_ j$ are smooth as well. $\square$