Theorem 7.7.49181532558D7

Proof. After an affine change of coordinates $z\mapsto z_0+Rz$ with $R>0$, we may assume that $B$ is the unit ball; plurisubharmonicity, the boundary conditions, $C^2$ data and the vanishing of the Monge–Ampère measure are preserved by such a change, by proposition 5.9.3 and proposition 7.2.2(5). We call the functions $v$ in the statement the subsolutions for $\varphi $ of the homogeneous equation. If $v$ is a subsolution for $\varphi $ and $\varphi \le \varphi '+c$ on $\partial B$ with a constant $c$, then $v-c$ is a subsolution for $\varphi '$. Every subsolution is subharmonic on $B$ (theorem 5.9.1), so the comparison principle (theorem 3.4.5 and the remark following it), applied with the harmonic function $P\varphi $, the Poisson integral of $\varphi $ given by theorem 3.1.18, gives $v\le P\varphi $, and $P\varphi \le \max _{\partial B}\varphi $ because the Poisson kernel is positive with integral $1$. Hence, as soon as the class of subsolutions is nonempty, its upper envelope $H$ is defined, and $H\le P\varphi $.

Step 1: the envelope for $C^2$ data. Suppose that $\varphi $ is the restriction of a $C^2$ function on a neighborhood of $\partial B$. Multiplying this function by a cutoff equal to $1$ near $\partial B$, we obtain a $C^2$ function $\tilde\varphi $ with compact support in $\mathbb {C}^ n$ and $\tilde\varphi =\varphi $ on $\partial B$. Its complex Hessian is continuous with compact support, so there is $\lambda >0$ with $\sum _{j,k}\tilde\varphi _{j\bar k}\xi _ j\bar\xi _ k\ge -\lambda |\xi |^2$ everywhere. The function $s=\tilde\varphi +\lambda (|z|^2-1)$ is then of class $C^2$ with positive semidefinite complex Hessian, so it is plurisubharmonic by proposition 5.9.2. As $s=\varphi $ on $\partial B$, it is a subsolution. Hence $s\le H\le P\varphi $. By theorem 5.10.2, the upper semicontinuous regularization $H^*$ is plurisubharmonic, and $s\le H^*\le P\varphi $ since $P\varphi $ is continuous. As $s$ and $P\varphi $ are continuous on $\overline B$ with boundary values $\varphi $, we have $H^*(z)\to \varphi (x)$ as $z\to x\in \partial B$; so $H^*$ is a subsolution, $H^*\le H$, and $H=H^*$ is plurisubharmonic. Moreover $H(z)\to \varphi (x)$ as $z\to x\in \partial B$, and $H$ is bounded.

Step 2: second differences. Let $K\subset B$ be compact, and let $S_{a,h}$, $h_0$ and $C$ be as in lemma 7.7.1. For $x\in \partial B$ and $a\in K$, the function $g(h)=\tilde\varphi (S_{a,h}(x))$, $|h|\le h_0$, is of class $C^2$, and by the chain rule its second derivatives are bounded by a constant $2C_ K$ independent of $a$ and $x$, since the first and second derivatives of $\tilde\varphi $ are bounded and those of $S_{a,h}(x)$ with respect to $h$ are bounded by $C$. Since $g(0)=\varphi (x)$, Taylor's formula with integral remainder gives $g(h)+g(-h)-2g(0)=\int _0^1(1-t)\bigl (D^2g(th)[h,h]+D^2g(-th)[h,h]\bigr )\, \mathrm dt\le 2C_ K|h|^2$. As $S_{a,\pm h}$ maps $\partial B$ to $\partial B$, where $\tilde\varphi =\varphi $, this reads

The function $w=\frac12(H\circ S_{a,h}+H\circ S_{a,-h})-C_ K|h|^2$ is plurisubharmonic on $B$, by proposition 5.9.3, proposition 5.9.5, since $S_{a,\pm h}$ are automorphisms of $B$ and $H$ is bounded. If $y\in \partial B$ and $z\to y$ in $B$, then $S_{a,\pm h}(z)\to S_{a,\pm h}(y)\in \partial B$, so $H(S_{a,\pm h}(z))\to \varphi (S_{a,\pm h}(y))$ by Step 1, and the preceding inequality gives $\varlimsup _{z\to y}w(z)\le \varphi (y)$. Thus $w$ is a subsolution, so $w\le H$. Evaluating at $z=a$, where $S_{a,\pm h}(a)=a\pm h$, we obtain

Since $K$ was arbitrary and the points $a\pm h=S_{a,\pm h}(a)$ lie in $B$, lemma 7.7.2 applies on $U=B$. Thus $H$ is continuous on $B$ and belongs to $C^{1,1}_{\mathrm{loc}}(B)$, and its distributional real Hessian $A$ and complex Hessian $(H_{j\bar k})$ have the properties stated there: the second-order expansion holds almost everywhere, $(H_{j\bar k})$ is positive semidefinite almost everywhere, and $(\mathrm{dd}^{\mathrm{c}}H)^ n=n!\, \pi ^{-n}\det (H_{j\bar k})\, \mathrm dV_{2n}$.

Step 3: the equation. Suppose that $\det (H_{j\bar k})>0$ on a set of positive measure, and choose a point $a$ of this set at which $(H_{j\bar k}(a))$ is positive semidefinite and the second-order expansion holds. The Taylor polynomial $Q(z)=H(a)+\nabla H(a)\cdot (z-a)+\frac12(z-a)^ tA(a)(z-a)$ is a real quadratic polynomial with real Hessian $A(a)$, so its complex Hessian is the constant matrix $(H_{j\bar k}(a))$; this matrix is positive semidefinite with positive determinant, hence positive definite. Let $\eta >0$ be smaller than its smallest eigenvalue; then $Q-\eta |z-a|^2$ has positive semidefinite complex Hessian and is plurisubharmonic (proposition 5.9.2). For sufficiently small $r>0$ we have $\overline{B(a,r)}\subset B$ and, by the expansion, $|H-Q|<\eta r^2/2$ on the sphere $|z-a|=r$. The polynomial $\tilde Q=Q-\eta |z-a|^2+\eta r^2/4$ is then strictly smaller than $H$ on that sphere, since there $\tilde Q<H+\eta r^2/2-3\eta r^2/4$, and $\tilde Q(a)=H(a)+\eta r^2/4>H(a)$. By proposition 5.9.5, applied with $D=B(a,r)$ and $v=\tilde Q$, whose boundary inequality holds because $\tilde Q$ and $H$ are continuous, the function equal to $\max \{ H,\tilde Q\} $ on $B(a,r)$ and to $H$ elsewhere is plurisubharmonic on $B$. It coincides with $H$ near $\partial B$, so it is a subsolution, and it exceeds $H$ at $a$, a contradiction. Hence $\det (H_{j\bar k})=0$ almost everywhere, and $(\mathrm{dd}^{\mathrm{c}}H)^ n=0$. This proves the theorem for $C^2$ data, including the $C^{1,1}$ regularity, except for uniqueness.

Step 4: continuous data and uniqueness. For continuous $\varphi $, the function $z\mapsto \varphi (z/|z|)$ is continuous on the annulus $\{ 1/2<|z|<2\} $, and its mollifications $\varphi _ j$, $j\ge 4$, are smooth on the neighborhood $\{ 3/4<|z|<3/2\} $ of $\partial B$ and converge to $\varphi $ uniformly on $\partial B$ (lemma 2.5.1(1)). Let $H_ j$ be the envelope for $\varphi _ j|_{\partial B}$; by Steps 1–3 it is continuous on $\overline B$ with boundary values $\varphi _ j$, plurisubharmonic, and $(\mathrm{dd}^{\mathrm{c}}H_ j)^ n=0$. Put $c_ j=\max _{\partial B}|\varphi _ j-\varphi |$. Since $H_ j-c_ j$ is a subsolution for $\varphi $, the class of subsolutions for $\varphi $ is nonempty, and its envelope $H$ satisfies $H_ j-c_ j\le H$; symmetrically, $H-c_ j\le H_ j$. Therefore $\sup _ B|H_ j-H|\le c_ j\to 0$, so $H$ is a uniform limit of continuous functions on $B$; it is continuous, extends continuously to $\overline B$ with the boundary values $\varphi $, and is plurisubharmonic, being equal to its regularization $H^*$, which is plurisubharmonic by theorem 5.10.2. By proposition 7.2.6, $(\mathrm{dd}^{\mathrm{c}}H)^ n=\lim _ j(\mathrm{dd}^{\mathrm{c}}H_ j)^ n=0$. Finally, if $H'$ and $H''$ are two solutions, then $H'-H''\to 0$ at $\partial B$, and the last assertion of theorem 7.3.11, applied in both directions, gives $H'=H''$. $\square$