Pome

Proposition 15.18.3D2F8F8FD5A17

Proposition 15.18.3.D2F8F8FD5A17 Let $D\Subset \mathbb C^ n$ be a bounded domain and let $\vartheta $ be a smooth real $(1,1)$-form near $\overline D$. Suppose $u,v\in C(\overline D)$ are $\vartheta $-plurisubharmonic,

\[ (\vartheta +\mathrm{dd}^{\mathrm{c}}u)^ n\geq F\beta ^ n, \qquad (\vartheta +\mathrm{dd}^{\mathrm{c}}v)^ n\leq F\beta ^ n, \]

where $F>0$ is continuous on $D$. If $u\leq v$ on $\partial D$, then $u\leq v$ in $D$.

Proof. Step 1: $v$ admits no strict smooth lower test. Suppose, to the contrary, that a $C^2$ function $q$ touches $v$ from below at a point $p\in D$ and that, at $p$,

\[ \vartheta +\mathrm{dd}^{\mathrm{c}}q>0, \qquad \det _\beta (\vartheta +\mathrm{dd}^{\mathrm{c}}q)>F. \]

Choose a sufficiently small $a>0$. After replacing $q$ by $q-a|z-p|^2$, which makes the contact strict, and restricting to a small ball $B_ r(p)$, there are fixed $\lambda ,\eta >0$ such that

\[ A:=\vartheta +\mathrm{dd}^{\mathrm{c}}q\geq \lambda \beta , \qquad A^ n\geq (F+\eta )\beta ^ n. \]

To compare with closed currents, freeze the background at $p$: put $\kappa =\sup _{B_ r}\lVert \vartheta -\vartheta (p)\rVert _\beta $ and $B_0=\vartheta (p)+\kappa \beta $. Then $S=B_0-\vartheta $ satisfies $0\leq S\leq 2\kappa \beta $. For small $s>0$ the set $E=\{ v<q+s\} $ is nonempty and relatively compact in $B_ r$. If $Q_0$ is a quadratic potential for the constant form $B_0$, then

\[ P=\mathrm{dd}^{\mathrm{c}}(v+Q_0)=\vartheta +\mathrm{dd}^{\mathrm{c}}v+S, \qquad Q=\mathrm{dd}^{\mathrm{c}}(q+s+Q_0)=A+S\geq \lambda \beta \]

are closed positive Hessians. Applying (5.27.7), expanding $P$, and using $S\leq 2\kappa \beta \leq 2\kappa \lambda ^{-1}Q$, we obtain

\[ \int _ E P^ n\leq \int _ E F\beta ^ n+e_\kappa \int _ E P^ n, \qquad e_\kappa =\left(1+\frac{2\kappa }{\lambda }\right)^ n-1. \]

On the other hand, (5.27.7) with $j=n$ gives

\[ \int _ E P^ n\geq \int _ E Q^ n\geq \int _ E(F+\eta )\beta ^ n. \]

If $M$ bounds $F$ on the ball, the two estimates give $(1-e_\kappa )\eta \leq e_\kappa M$. Since $\kappa \to 0$, and hence $e_\kappa \to 0$, as $r\downarrow 0$, this is impossible for small $r$. Thus no such lower test exists.

Step 2: comparison.

Suppose that $\max _ D(u-v)>0$. We regularize $u$ to produce a forbidden lower test for $v$. Let $\chi _\delta $ be a nonnegative smooth mollifier of mass one and radius $\delta >0$. For small $t>0$ and $C\geq \sup _ D|z|^2$, set

\[ u_ t=u+t(|z|^2-C). \]

Then $u_ t\leq v$ on $\partial D$, the positive maximum persists, and $T_ t=\vartheta +\mathrm{dd}^{\mathrm{c}}u_ t\geq t\beta $. Lemma 5.28.7 gives

\[ T_ t^ n\geq (F^{1/n}+t)^ n\beta ^ n. \]

Choose a relatively compact neighborhood $W$ of the positive maximum set. By lemma 5.28.10,

\[ (T_ t*\chi _\delta )^ n \geq \bigl ((F^{1/n}+t)*\chi _\delta \bigr )^ n\beta ^ n. \]

Since

\[ \vartheta +\mathrm{dd}^{\mathrm{c}}(u_ t*\chi _\delta ) =T_ t*\chi _\delta +(\vartheta -\vartheta *\chi _\delta ), \]

the convolution satisfies $T_ t*\chi _\delta \geq t\beta $. If $\epsilon _\delta =\| \vartheta -\vartheta *\chi _\delta \| _{C^0(W)}$, then

\[ \vartheta +\mathrm{dd}^{\mathrm{c}}(u_ t*\chi _\delta ) \geq (1-\epsilon _\delta /t)(T_ t*\chi _\delta ). \]

For fixed $t>0$, uniform convergence of the convolved density and $\epsilon _\delta \to 0$ show that the left-hand side is positive and has determinant strictly larger than $F$ on $W$ for small $\delta $. Since $u_ t*\chi _\delta \to u_ t$ uniformly on $W$, the difference $u_ t*\chi _\delta -v$ still attains its maximum over $\overline W$ at an interior point of $W$ for small $\delta $. Subtracting this maximum from $u_ t*\chi _\delta $ produces a smooth function touching $v$ from below with the properties excluded in Step 1, a contradiction. $\square$

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