8.10 Sobolev capacity and two Hessian factorsFB2B0B645AB5

Proof. For open $G_ i$ choose Sobolev functions $0\leq f_ i\leq 1$ equal to one almost everywhere on $G_ i$, with a summably small excess in their squared norms over the capacity infima. The Sobolev chain rule gives

Indeed, almost everywhere the gradient of the maximum is the gradient of an active function, and the squared value is bounded by the sum of the squared values. If the sum is finite, these maxima converge in $L^2$, are bounded in $H^1$, and have a weak $H^1$ limit equal to one almost everywhere on $\bigcup _ iG_ i$. Lower semicontinuity proves subadditivity for open sets. Taking open supersets proves it for arbitrary sets. If the sum is infinite there is nothing to prove. The level-set estimate follows by testing with $\min \{ 1,|f|/a\} $ and the same Sobolev chain rule.

Fix a compact $K\subset G$ and a Sobolev competitor $0\leq f\leq 1$ for the open set $G$. Positive local mollification and a partition of unity give smooth nonnegative $f_\epsilon \to f$ in $H^1$ that satisfy $f_\epsilon \geq 1$ on $K$ for small $\epsilon $: every convolution ball meeting $K$ then lies in $G$, where $f=1$ almost everywhere. The assumed estimate gives $\mu (K)\leq C\lim _\epsilon \| f_\epsilon \| _{H^1}=C\| f\| _{H^1}$. Take the infimum over $f$ and then the supremum over compact $K\subset G$ to obtain the estimate for $G$. Outer regularity and open supersets give it for every Borel $E$. $\square$

Proof. First let $u$ be smooth admissible. The first-cone inequality gives $\mu _ u\geq 0$, since $T$ is conformal to $\omega ^{n-1}$, and $\mathrm{dd}^{\mathrm{c}}T=0$ gives $\mu _ u(X)=A$. Applying distributional Stokes to $\mathrm{dd}^{\mathrm{c}}(u^2)$ yields the energy identity

Together with the $L^2$ bound this controls $\| u\| _{H^1}$. For bounded admissible $u$, choose smooth admissible $u_ j\downarrow u$ by 8.6.6, discard a finite initial segment, and take $\| u_ j\| _\infty \leq M+1$ as in 8.9.1. Bounded convergence gives strong $L^2$ convergence. The energy bound gives weak $H^1$ compactness, and every weak cluster point agrees with $u$ in $L^2$; thus the entire sequence converges weakly to $u$ in $H^1$. This proves the Sobolev bound. The trace measures converge weakly to $\mu _ u$, preserving positivity and total mass.

For any smooth real $\phi $, the first-order Stokes identity is

It remains valid for $u\in H^1$ by smooth Sobolev approximation. Both terms are bounded by $C\| u\| _{H^1}\| \phi \| _{H^1}$; the background term $\int \phi \omega \wedge T$ is bounded by $C\| \phi \| _{H^1}$. The preceding capacity lemma gives the bound for $\mu _ u$. In particular the derivative $\mathrm dT$ has not been discarded.

For a smooth admissible approximant, write the eigenvalues of $\omega +\mathrm{dd}^{\mathrm{c}}u_ j$ in an $\omega $-unitary frame as $\lambda _ i$. Since $m\geq 2$,

Consequently every coefficient of this form is dominated by its trace. On a fixed coordinate chart this bounds the coefficient variations of $\mathrm{dd}^{\mathrm{c}}u_ j$ by $C(\mu _{u_ j}+\omega ^ n)$. Passing to the weak measure limits of 8.9.1 gives the same domination by $C(\mu _ u+\omega ^ n)$. Finally $\int _ E\omega ^ n\leq \operatorname {Cap}_1(E)$ and $\operatorname {Cap}_1(E)\leq \| 1\| _{H^1}^2$ absorb the smooth term into the claimed square-root estimate. $\square$

Proof. Choose smooth admissible $u_ j\downarrow u$. By the preceding lemma these functions converge weakly to $u$ in $H^1$. Mazur's lemma gives finite convex combinations $w_ N$ of terms with $j\geq N$ such that

Because $u\leq w_ N\leq u_ N$, they converge pointwise to the canonical function $u$, not just to an unspecified almost-everywhere representative. Put

The capacity tools imply

Outside $G_{N_0}$, the successive differences form a uniformly summable series. Thus $w_ N$ converge uniformly there and their limit $u$ is continuous on that closed set. For a countable family, choose the exceptional open sets with capacities less than $\epsilon 2^{-i}$ and take their union. Countable subadditivity proves the last assertion. $\square$

Proof. Bounded convergence gives $u_ j\to u$ in $L^1$. Their complex Hessians therefore converge in distributions. The preceding coefficient bounds are uniform, so this is weak convergence of each signed coefficient measure against continuous compactly supported tests. The trace measures also converge weakly. Every one of these measures has total variation bounded by $C_ M\operatorname {Cap}_1(E)^{1/2}$ on a Borel set $E$.

Fix $\epsilon >0$. Quasicontinuity supplies an open $G$ with $\operatorname {Cap}_1(G)<\epsilon $ on whose complement $v$ and every $v_ j$ are continuous. Dini's theorem on the compact set $X\setminus G$ gives uniform convergence there. Extend $v|_{X\setminus G}$ to a continuous function $F$ on $X$ with the same uniform bound, by Tietze's theorem. For any one of the signed coefficient measures $\nu _ j\to \nu $ and a fixed continuous compactly supported test $\psi $, subtract $F\nu _ j$ and $F\nu $. The resulting error is at most

The term with $F$ converges by weak measure convergence. First let $j\to \infty $ and then $\epsilon \downarrow 0$. This proves the assertion for the Hessian coefficients and for the positive trace measures. $\square$

Proof. For smooth admissible approximations $u_ k\downarrow u$ and $v_ k\downarrow v$, the weighted convergence proposition gives

The smooth current on the left is $\mathrm{dd}^{\mathrm{c}}v_ k\wedge \mathrm{dd}^{\mathrm{c}}u_ k$ and is symmetric. Applying the same argument in the opposite order proves $D_2(u,v)=D_2(v,u)$. For the bounded sequences in the statement, weighted convergence followed by $\mathrm{dd}^{\mathrm{c}}$ gives the asserted simultaneous distributional limit.

The displayed subset expansion is the current of 8.9.6 with $p=2$. Its lower-degree terms converge weakly, so its whole distributional limit is $Q_2(u,v)$. Positivity and the uniform local mass bounds turn this into weak convergence of currents of order zero. When $m>2$, the same conclusion for $D_2$ follows from the uniform coefficient bounds below degree $m$ in 8.9.3. $\square$

Proof. It suffices to treat a compact subset of a submanifold chart, since finitely many such subsets cover $E$ and the capacity is subadditive. Choose real coordinates $(s,t)$ there, with $t\in \mathbb R^2$ and the submanifold given by $t=0$. Choose $0\leq \eta \leq 1$ smooth, supported in this chart and equal to one near the compact subset. Put $r=|t|$. For $0<\epsilon <1/2$ define

The Sobolev function $f_\epsilon =\eta h_\epsilon $ equals one on an open neighborhood of that subset. On the fixed compact coordinate set the Sobolev norm is equivalent to the Euclidean one. Integration in the two normal variables gives

The derivatives of $\eta $ contribute at most $C\epsilon ^2$. Taking the infimum over open neighborhoods proves the assertion. $\square$

Proof. Choose holomorphic coordinates $z_ j=x_ j+\mathrm i y_ j$ and a smooth nonnegative cutoff $\chi $ supported in the chart and equal to one on a smaller coordinate box. Extend

by zero outside the chart. These functions are globally continuous and bounded. To check their curvature, replace $|x_ j|$ by $g_\delta (x_ j)=(x_ j^2+\delta ^2)^{1/2}$, with $0<\delta \leq 1$. The Hessian of $\chi g_\delta $ has the nonnegative term $\chi \mathrm{dd}^{\mathrm{c}}g_\delta $; all its remaining terms are uniformly bounded forms, since $g_\delta $ and $g_\delta '$ are uniformly bounded on the fixed support. Thus $\mathrm{dd}^{\mathrm{c}}(\chi g_\delta )\geq -C\omega $ with $C$ independent of $\delta $. Choosing $a>0$ sufficiently small gives $\omega +\mathrm{dd}^{\mathrm{c}}(a\chi g_\delta )\geq \omega /2$ for both coordinates. Their decreasing limits $u,v$ are therefore $\omega $-plurisubharmonic and admissible.

On the box where $\chi =1$, set $\alpha _ j=\frac{\mathrm i}{4\pi }\mathrm dz_ j\wedge \mathrm d\bar z_ j$. The book's normalization gives

The measures $g_\delta ^{\prime \prime }(t)\, \mathrm dt$ have mass two, while their mass outside any fixed neighborhood of zero tends to zero. The two smooth factors depend on distinct real variables, so their product is an approximate identity in $(x_1,x_2)$ and converges to the tensor-product measure. By 8.10.5, locally

Choose $E$ to be a compact piece of $\{ x_1=x_2=0\} $ inside this box with nonempty relative interior. The preceding lemma gives $\operatorname {Cap}_1(E)=0$. On this box, the positive measure $D_2(u,v)\wedge \omega ^{n-2}$ has strictly positive mass on $E$, since $\omega $ is strictly positive. The smooth term $\omega ^ n$ and the two single-Hessian terms in the expansion

have zero mass on $E$: each single-Hessian term is absolutely continuous in the other normal real variable. Hence the asserted mass is strictly positive. A bound by $C\operatorname {Cap}_1(F)^ b$ for all Borel $F$, with any $b>0$, would contradict this choice of $E$. $\square$