Pome7A246CDE0880The comparison principle below, for continuous sub- and supersolutions on a bounded domain, identifies the given continuous potential with limits of smooth Dirichlet solutions. Since the background need not be closed, the classical Bedford–Taylor comparison does not apply directly. We freeze the background at a point, apply the closed-current comparison formula (5.27.7) to quadratic modifications, and control the error by the mixed inequalities of lemma 5.28.7, lemma 5.28.10.
The local arguments of this section take place in coordinate balls, always with compact closure in a larger coordinate chart. There we put $\beta =\mathrm{dd}^{\mathrm{c}}|z|^2$, write $\partial _ i=\partial /\partial z_ i$ and $\partial _{\bar i}=\partial /\partial \bar z_ i$, and take coefficient matrices of real $(1,1)$-forms relative to $\beta $. For such a form $A$, $\operatorname {tr}_\beta A$ and $\det _\beta A$ denote the trace and the determinant of its coefficient matrix, so that $A^ n=(\det _\beta A)\beta ^ n$, and $\Delta _\beta =\operatorname {tr}_\beta \mathrm{dd}^{\mathrm{c}}$. For a positive form $T$, $\operatorname {tr}_ T$ is the trace computed with the inverse of the coefficient matrix of $T$. Indices run from $1$ to $n$, repeated indices in tensor expressions are summed, and $I$ denotes the identity matrix. In local statements, $\vartheta $ denotes a smooth real $(1,1)$-form, not necessarily closed, which plays the role of $\theta $.
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,
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 $p$,
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
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
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
On the other hand, (5.27.7) with $j=n$ gives
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
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
Choose a relatively compact neighborhood $W$ of the positive maximum set. By lemma 5.28.10,
Since
the convolution satisfies $T_ t*\chi _\delta \geq t\beta $. If $\epsilon _\delta =\| \vartheta -\vartheta *\chi _\delta \| _{C^0(W)}$, then
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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