15.18 Interior smoothness on the regular locus899B82DE5E84

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. $\square$

Proof. Choose $C$ so that $v=u+C|z|^2$ is plurisubharmonic, and use nonnegative radial mollifiers for which $v_\eta \downarrow v$. The smooth matrix convolutions $T_\eta =T*\chi _\eta $ can be written as

Bedford–Taylor monotone continuity and smooth convergence of the background imply $T_\eta ^ k\to T^ k$ for every $k$.

(2)$\Rightarrow $(3). Assume (2). For every Hermitian matrix $M\geq \delta I$,

Apply this to the regularizations and pass to currents. Equation (15.18.6) gives

which is (3).

(3)$\Rightarrow $(1). Assume (3), say $\operatorname {tr}_\beta T\leq L$. Positivity makes every coefficient of the matrix-valued current $T$ a bounded density. Hence the Euclidean Laplacian of $u$ is locally bounded, and interior Poisson estimates give $u\in W^{2,q}_{\mathrm{loc}}$ for every finite $q$. Mollification converges in $W^{2,q}$; taking $q>2n$ identifies the bounded-potential determinant with the ordinary almost-everywhere determinant. Thus $\det _\beta T=F$ almost everywhere. On a smaller ball with $F\geq m>0$, all eigenvalues of $T$ are at most $L$ and their product is at least $m$, so

almost everywhere.

It remains to show that this strong solution is smooth. Let $P$ be the real linear map from real symmetric $2n\times 2n$ Hessian matrices to Hermitian $n\times n$ matrices, sending $D^2u$ to the coefficient matrix of $\mathrm{dd}^{\mathrm{c}}u$ relative to $\beta $, and let $\Theta (x)$ be the matrix of $\vartheta $. With $\delta =mL^{1-n}$, define the concave extension

For fixed $A$, write $\mathcal E_ A(x,M)$ for the expression in braces minus $\log F(x)$. The tangent-plane formula for $\log \det $ gives

The operator $G$ is concave and uniformly elliptic on real symmetric matrices. The affine operators $\mathcal E_ A$ have constant principal matrices with eigenvalues bounded above and below in terms of $\delta $ and $L$. Their $x$-dependence is smooth uniformly in $A$, so $G$ has a uniform local Lipschitz modulus in $x$.

We claim that $u$ is a viscosity solution of $G(x,D^2u)=0$. Suppose that a smooth function $\psi $ touches $u$ from above at $x_0$ and that $G(x_0,D^2\psi )<0$. Some fixed matrix $A$ then satisfies $\mathcal E_ A(x,D^2\psi )<-\eta $ on a small ball. Since $G(x,D^2u)=0$ almost everywhere, $\mathcal E_ A(x,D^2u)\geq 0$ there. Thus $u-\psi $ has an interior maximum and satisfies

After a small quadratic perturbation makes the contact strict, this contradicts the Aleksandrov maximum principle [Chapter 9, Theorem 9.1, GT01]. If instead $\psi $ touches $u$ from below and $G(x_0,D^2\psi )>0$, then $\mathcal E_ A(x,D^2\psi )>\eta $ for every admissible $A$ on a small ball. At almost every point choose an admissible $A(x)$ with $\mathcal E_{A(x)}(x,D^2u)<\eta /2$. A measurable choice is obtained by taking the first qualifying element of a countable dense family. The function $\psi -u$ then violates the same maximum principle for the measurable uniformly elliptic coefficients $A(x)^{-1}$. This proves the claim.

The classical variable-coefficient Evans–Krylov theorem gives $u\in C^{2,\alpha }_{\mathrm{loc}}$ for some $0<\alpha <1$. The original equation is now uniformly elliptic with smooth data. Difference quotients satisfy linear equations whose principal coefficients are averages of inverse matrices along positive definite segments. Interior Schauder estimates first give $u\in C^{3,\alpha }_{\mathrm{loc}}$ and then, by iteration, $u\in C^\infty $. This proves (3)$\Rightarrow $(1).

(1)$\Rightarrow $(2). If $u$ is smooth, then $T$ is a smooth semipositive form with determinant $F>0$, so it is positive definite; on a smaller neighborhood this gives (2). $\square$

Proof. This is the maximum-principle computation behind the classical second-order estimate. The non-closedness of $\vartheta $ produces a first-order error term, which is absorbed by the convex term $\mathrm{e}^{-v}$.

At the point in question make a constant unitary change of coordinates so that the matrix of $T$ is diagonal, with eigenvalues $\lambda _ i$. Write

Differentiating $\log \det T=\log F$ and commuting derivatives gives

The non-closedness of $\vartheta $ enters through the bounded vector

Since $\partial _ pS-E_ p=\sum _ k\partial _ kT_{p\bar k}$, weighted Cauchy–Schwarz gives $|\partial S-E|_ T^2\leq SP$. From (15.18.10),

Set $g_ A(s)=-As+\mathrm{e}^{-s}$. At a local maximum of $Q_ A$, $\partial \log S=-g_ A'(v)\partial v$ and $\Delta _ Tv=n-\operatorname {tr}_ TR$. Combining these identities with (15.18.11), then completing the square in $\partial v$, yields

Here $|E|_ T^2\leq CH$ and the bounds for $v$ have been used. If $S\geq K\sqrt A$ and $K$ is large, the bracket is at least $\varepsilon A/2$. Thus $H\leq C$ independently of $A$ and $u$. The determinant upper bound then gives $S\leq C'$. Enlarging $K$ makes this incompatible with $S\geq K\sqrt A$, proving (15.18.9). $\square$

Proof. We follow the classical boundary estimates for the Dirichlet problem on a ball and keep track of the dependence on $L=1+\lVert g\rVert _{C^4}$. Barriers bound the first derivatives, and differentiation along rotations bounds the mixed normal–tangential second derivatives on the boundary. A lower bound for the tangential block of $T$ then bounds the double normal derivative through the equation, and lemma 15.18.8 reduces the interior bound to the boundary bound.

Step 1: first derivatives. Fix a sufficiently large constant $K>0$, depending only on the fixed data. Extend $g$ to $G$ on $\overline B$ with $\lVert G\rVert _{C^4}\leq CL$. For $A=CL$ large enough,

Proposition 15.18.3 gives $\underline u\leq u$. If $h$ solves $\Delta _\beta h=-C_\vartheta $ with $h=g$ on $\partial B$ and $C_\vartheta \geq \sup \operatorname {tr}_\beta \vartheta $, then $u\leq h$. Boundary differentiation of these barriers gives

Let $H=\operatorname {tr}_ T\beta $, $\mathcal L=\operatorname {tr}_ T\mathrm{dd}^{\mathrm{c}}$, and $w=u-\underline u\geq 0$, the linearized operator and the barrier function used below. Taking $K$ fixed and sufficiently large gives

Let $E=\sum _ k(z_ k\partial _{z_ k}+\bar z_ k\partial _{\bar z_ k})$ be the real radial field. Write $\Theta $ for the coefficient matrix of $\vartheta $. Differentiating $\log \det T=\log F$ and commuting $E$ with complex second derivatives yields

Hence $|\mathcal L(Eu)|\leq C(1+H)$. The same estimate holds for $Zu$ when $Z$ is an infinitesimal unitary rotation. Applying the minimum principle to $CL+C'w\pm Yu$, with $Y=E$ or $Z$, gives

Step 2: mixed second derivatives on the boundary. For a rotation $Z$, the function $R_ Z=Z(u-G)$ vanishes on $\partial B$ and satisfies $|\mathcal LR_ Z|\leq CL(1+H)$. The barriers $CLw\pm R_ Z$ and (15.18.17) imply $|\partial _\nu R_ Z|\leq CL^2$ on the boundary. Rotations span the real tangent space of $\partial B$, so all real mixed normal–tangential second derivatives are $O(L^2)$. Pure tangential second derivatives are $O(L)$ by differentiating $u=g$ twice and using (15.18.16). Thus all complex Hessian entries except the complex normal–normal entry are $O(L^2)$.

Step 3: the double normal derivative. It remains to bound the tangential block of $T$ from below. For $n\geq 2$, let

Choose a minimizing pair $(p,\xi )$ and extend $\xi $ to a smooth unit tangential field near $p$, with uniformly bounded derivatives. On the sphere, any smooth function vanishing there satisfies

Define

Then $P\geq 0$ on the nearby spherical boundary and $P(p)=0$. The boundary first-derivative bound also gives $\lambda _0\leq CL$ by the tangential identity above. Consequently

On a fixed cap $D=B\cap B_ r(p)$ consider

Choose $C_2$ so that $\Psi \geq 0$ on the interior spherical part of $\partial D$, and then $C_1$ so that $\mathcal L\Psi \leq 0$ by (15.18.17). The minimum principle yields $\Psi \geq 0$. At $p$ the normal derivative of $|z-p|^2$ vanishes, and therefore

Equations (15.18.16)–(15.18.18) now give $D^2u(\nu ,\nu )(p)\leq CL^2$, hence $\operatorname {tr}_\beta T(p)\leq CL^2$. Positivity and $\det _\beta T=F$ imply

At any boundary point write

in tangential–normal blocks, with $B_0$ an $(n-1)\times (n-1)$ Hermitian matrix, $b\in \mathbb C^{n-1}$ and $d\in \mathbb R$. We have $|b|\leq CL^2$ and $B_0\geq \lambda _0I$. The determinant equation and the Schur complement give

which is polynomial in $L$. For $n=1$, the trace is $F$ and this argument is unnecessary.

Step 4: the interior. Choose a fixed quadratic $q$ with $\vartheta +\mathrm{dd}^{\mathrm{c}}q\geq \varepsilon \beta $. Apply lemma 15.18.8 with one fixed $A$. At an interior maximum of its auxiliary function the trace is uniformly bounded, while at a boundary maximum the preceding estimate applies. Since the remaining terms are uniformly bounded, (15.18.15) follows. $\square$

Proof. Since $\varphi $ is Hölder continuous near $\overline B$, mollification at scale $1/j$ gives smooth boundary approximations $g_ j$ with

Choose a fixed quadratic $q$ with $\eta =\theta +\mathrm{dd}^{\mathrm{c}}q>0$ near $\overline B$. Extend $g_ j$ smoothly to $G_ j$. For a sufficiently large, polynomially bounded $A_ j$, the function

has boundary value $g_ j-q$ and satisfies

Apply the smooth Dirichlet theorem recalled at the beginning of this subsection, with density $H=F\beta ^ n/\eta ^ n$ to obtain a solution $w_ j$ for the Hermitian background $\eta $; put $u_ j=w_ j+q$. Comparing $u_ j$ with $\varphi \pm \lVert g_ j-\varphi \rVert _\infty $ by proposition 15.18.3 proves the first bound in (15.18.23). The functions $u_ j$ are therefore uniformly bounded. Proposition 15.18.14 and the polynomial $C^4$ boundary norm give the second bound. $\square$

Proof. Step 1: the trace bound (15.18.30). Write $e_ j=Cj^{-a}$, fix $d=\kappa R^2/2$, and choose $D>(b+1)/d$. Choose $\eta >0$ so that $2D\eta <\gamma /2$. For small $\delta >0$, the neighborhood

contains $0$, has $v\geq m-\eta $ on its closure, and satisfies $v\geq m+d$ on $\partial B\cap \overline W_\delta $ with the fixed $d$ above. These properties follow from (15.18.26), continuity, and compactness. The remaining part of the boundary of $W_\delta $ is a compact subset of $\overline B\setminus \mathcal S$, where (15.18.29) applies.

Set $A_ j=D\log j$. Apply the maximum principle on $\overline W_\delta $ to

At an interior maximum, lemma 15.18.8 gives $\log \operatorname {tr}_\beta (\vartheta +\mathrm{dd}^{\mathrm{c}}u_ j)\leq C+\frac12\log A_ j$. On the part of $\partial W_\delta $ inside $B$, which stays away from $\mathcal S$, the trace has a bound independent of $j$. On the outer part $\partial B\cap \overline W_\delta $, (15.18.28) and the gap $d$ give

and the choice $Dd>b+1$ makes this smaller than the bound below for large $j$. Hence throughout $W_\delta $,

On a small ball about $0$ where $v\leq m+\eta $, this implies

By the choice of $\eta $, the logarithmic factor and $A_ je_ j$ are absorbed into the remaining power. This proves (15.18.30).

Step 2: $C^{1,\alpha }$ convergence. On a smaller ball, the real Laplacian of $u_ j$ is bounded by $Cj^\gamma $. Interior Poisson estimates give a $W^{2,q_0}$ bound for every finite $q_0$, and Sobolev embedding with $q_0$ large then gives, for every $\sigma <1$,

For $d_ k=u_{2^{k+1}}-u_{2^ k}$,

If $\alpha <\sigma $ and $t=(1+\alpha )/(1+\sigma )$, Hölder interpolation gives

Since $\gamma $ may be arbitrarily small, the exponent can be made negative. The dyadic series converges in $C^{1,\alpha }$ to its uniform limit $u$. If $2^ k\leq j<2^{k+1}$, interpolation applied to $u_ j-u_{2^ k}$ gives the same decaying bound. Hence the full sequence converges to $u$ in $C^{1,\alpha }$ on a smaller ball. $\square$

Proof. Where $u$ is smooth and strictly $\vartheta $-plurisubharmonic, it provides its own quadratic separation, so the argument of proposition 15.18.25 applies around every point of $O$. A first application gives trace growth of order $\sqrt{\log j}$; the resulting gradient convergence then allows the variant of lemma 15.18.8 with a gradient bound, which removes this growth.

For $p$ in a fixed compact subset of $O$, take a uniformly sized ball $B_ r(p)\Subset O$ and set

with $\kappa >0$ small, so $\vartheta +\mathrm{dd}^{\mathrm{c}}q_ p\geq \varepsilon \beta $. If $e_ j=Cj^{-a}$ and $v_{j,p}=u_ j-q_ p$, then

With $A_ j=D\log j$ and $D\kappa r^2>b+1$, the same maximum argument as above gives

The constants are uniform on compact subsets of $O$.

Put $T_ j=\vartheta +\mathrm{dd}^{\mathrm{c}}u_ j$ and $w_ j=u_ j-u$. On slightly larger compact sets, (15.18.33) bounds $|\Delta w_ j|$ by $C\sqrt{\log j}$. Poisson estimates and Sobolev embedding give

Interpolating with $\lVert w_ j\rVert _\infty \leq Cj^{-a}$ shows $\nabla w_ j\to 0$ uniformly on smaller compact sets. The gradients of $v_{j,p}$ are therefore uniformly bounded. Repeat the maximum argument with $\log \operatorname {tr}_\beta T_ j-A_ jv_{j,p}$, now using the variant of lemma 15.18.8 with a gradient bound. Its interior trace bound is now independent of $A_ j$, while the boundary term is suppressed as before. Evaluation at $p$ gives

proving (15.18.32). $\square$

Proof. The quantity $\mathcal R$ below is the squared norm of the difference between the Chern connection of $g$ and the flat connection. We derive a differential inequality for $\mathcal R$ in which the torsion enters through fixed derivatives of $\vartheta $, and combine it with the trace in a maximum principle; every constant is kept a power of $M$.

We compare the flat connection of $\beta $ with the Chern connection $\nabla $ of $g$. Write $g^{k\bar\ell }$ for the coefficients of the inverse matrix and $\bar\nabla $ for covariant differentiation in $(0,1)$ directions; tensor norms and inner products without subscript are computed with $g$. The difference of the two connections and its squared norm are

The torsion of $\nabla $ is determined by the fixed background, because the third derivatives of $u$ cancel in it. Its lowered and raised components are

Consequently, for a fixed integer $N$,

Every factor here is either a fixed derivative of $\vartheta $, a connection factor $\Psi $, or a factor of $g^{\pm 1}$; this explains the polynomial dependence.

For tensor fields, use $\Delta _ g=g^{p\bar q}\nabla _ p\nabla _{\bar q}$. We need the following curvature identities. With

set

The two curvature interchanges are

They follow by differentiating the torsion identity and its conjugate. Successive interchange and contraction give

Metric compatibility, (15.18.39), and the fixed smooth formula for $\operatorname {Ric}^{(1)}$ imply

Moreover,

and the Chern–Bianchi identity, including its torsion term, is

Consequently,

If $A^ k{}_ l=(\operatorname {Ric}^{(2)})^ k{}_ l$, define its action on a tensor of type $(1,2)$ by

Differentiating the squared norm and using metric compatibility gives

There is no mixed torsion term because the Chern torsion has type $(2,0)$. Using (15.18.39), (15.18.41), and (15.18.42) in (15.18.43), Young's inequality absorbs the products of $\sqrt{\mathcal R}$ with $|\nabla \Psi |+|\bar\nabla \Psi |$. We obtain

Let $s=\operatorname {tr}_\beta g$. Direct differentiation as in (15.18.10) gives

Take a cutoff $\zeta $ supported in $B$, equal to one on $B_{1/2}$, set $\eta =\zeta ^2$ and $h=2nM-s$, so $nM\leq h\leq 2nM$, and maximize

At a positive interior maximum of $Q$, the identity $\partial \mathcal R/\mathcal R=\partial h/h-\partial \eta /\eta $ gives

We have $|\partial \eta |_ g^2\leq CM\eta $, $|\Delta _ g\eta |\leq CM$ and $|\partial h|_ g\leq \sqrt n M\sqrt{\mathcal R}$. Insert (15.18.44) and (15.18.45) and multiply by $\eta Q$. The positive term is $M^{-1}Q^2$. The two terms involving $\sqrt{\mathcal R}$ are bounded in absolute value by a fixed power of $M$ times $Q^{3/2}$, since $\mathcal R=hQ/\eta $ and $h\leq 2nM$. The remaining cutoff terms are bounded by $CM^ N(Q+1)$. Therefore

Thus $Q$, and then $\mathcal R$ on $B_{1/2}$, are bounded by a fixed polynomial in $M$. Since $|\partial g|_\beta ^2\leq M^3\mathcal R$, this proves (15.18.36).

If $s\leq L$, the determinant lower bound gives $\lambda _{\min }(g)\geq (\inf F)L^{1-n}$, so $M$ is a fixed power of $1+L$. For a ball $B_ r(p)$, set $u_ r(z)=r^{-2}u(p+rz)$ and use the coefficient matrices $\vartheta _ r(z)=\vartheta (p+rz)$, $g_ r(z)=g(p+rz)$ and $F_ r(z)=F(p+rz)$. Then $g_ r=\vartheta _ r+\mathrm{dd}^{\mathrm{c}}u_ r$ in the unit coordinates and $\det g_ r=F_ r$. For $r\leq 1$ the derivatives of the fixed data remain uniformly bounded. The unit-ball estimate for the ordinary coefficient derivatives $\partial g_ r$ and their conjugates gives the factor $r^{-1}$ in (15.18.37). $\square$

Proof. Fix a center for concentric balls $B_ r\Subset B_0$, put $s_ j=\operatorname {tr}_\beta (\vartheta +\mathrm{dd}^{\mathrm{c}}u_ j)$, and set

For concentric balls $B_ r\Subset B_ R\Subset B_0$, $d=R-r\leq 1$, proposition 15.18.35 gives

Fix $p\in B_ r$ and let $\chi _ h$ be a nonnegative mollifier of radius $h\leq d/4$. Since

integration by parts, after subtracting the constant vector $\nabla u_ j(p)$, and the uniform $C^{1,\alpha }$ bound give

Equation (15.18.47) gives

Choose

Taking the supremum over $p\in B_ r$ and absorbing fixed powers of $d^{-1}$, we obtain

Choose $\alpha _*$ so close to one that $q<1$ for $\alpha >\alpha _*$. Choose $0<r_0<R_0$ with $B_{R_0}\Subset B_0$. Let $r_ k\uparrow R_0$ with $r_{k+1}-r_ k=(R_0-r_0)2^{-k-1}$. Iterating (15.18.48) gives

The infinite product converges because $\sum q^ k(1+k)<\infty $. For each fixed $j$, the final factor tends to one as $m\to \infty $. The resulting bound is independent of $j$. $\square$

of theorem 15.18.1. If $n=1$, the equation is the linear identity

Its right-hand side is smooth, so interior elliptic regularity gives $\varphi \in C^\infty (\Omega )$, and the original equation gives $\theta +\mathrm{dd}^{\mathrm{c}}\varphi =cf\, \omega >0$. We henceforth assume $n\geq 2$.

Let

This is closed in $\Omega $. Suppose $\mathcal S\neq \varnothing $. By lemma 5.40.7, the restriction of $\varphi -\rho $ to $\mathcal S$ attains its minimum $m$ at some $p\in \mathcal S$.

Choose nested coordinate balls

centered at $p$. On $B_0$, proposition 15.18.21 gives smooth exact Dirichlet solutions $u_ j$ converging uniformly to $\varphi $ at a polynomial rate and having polynomial global trace growth.

On the open set $\Omega \setminus \mathcal S$ the function $\varphi $ is smooth, and $\theta +\mathrm{dd}^{\mathrm{c}}\varphi $ is semipositive with positive determinant, hence positive definite. Proposition 15.18.31 therefore gives a uniform trace bound for the $u_ j$ on every compact subset of $B_0\setminus \mathcal S$.

In coordinates centered at $p$, take $\kappa >0$ small and set

After shrinking $B$, the lower bound $\theta +\mathrm{dd}^{\mathrm{c}}\rho \geq \omega $ gives $\theta +\mathrm{dd}^{\mathrm{c}}q>0$ on $\overline B$. If $v=\varphi -q$, then the minimizing property of $p$ gives

Thus proposition 15.18.25 applies on $B$ with the closed set $\mathcal S\cap \overline B$: the approximation rate and global polynomial trace bound come from proposition 15.18.21, and the bound away from $\mathcal S$ comes from proposition 15.18.31. Its second assertion gives, for every $0<\alpha <1$, a neighborhood of $p$ on which the approximants have a uniform $C^{1,\alpha }$ bound.

Choose $\alpha >\alpha _*$ as in proposition 15.18.46, and apply that proposition on a ball centered at $p$ inside this neighborhood. It yields

on a still smaller neighborhood of $p$. Since $u_ j\to \varphi $ uniformly, $\mathrm{dd}^{\mathrm{c}}u_ j\to \mathrm{dd}^{\mathrm{c}}\varphi $ as currents. The trace bound passes to the limit, so $\operatorname {tr}_\beta (\theta +\mathrm{dd}^{\mathrm{c}}\varphi )$ has a bounded density near $p$. Proposition 15.18.5 now implies that $\varphi $ is smooth near $p$, contradicting $p\in \mathcal S$.

Hence $\mathcal S=\varnothing $ and $\varphi \in C^\infty (\Omega )$. The form $\theta +\mathrm{dd}^{\mathrm{c}}\varphi $ is smooth and semipositive there, while

Its nonnegative eigenvalues have positive product, so every eigenvalue is positive. Therefore $\theta +\mathrm{dd}^{\mathrm{c}}\varphi >0$ on $\Omega $. $\square$

Proof. The analytic complement of $\Omega $ has zero mass for both measures, so it suffices to know the equation in the bounded-potential sense on $\Omega $. Apply theorem 15.17.1 with density $cf$ to obtain local Hölder continuity. All hypotheses of theorem 15.18.1 are now satisfied. $\square$