Proposition 15.18.57DF8FE8A18D7

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). $\square$