Theorem 7.9.2A092925717E1

Proof. For data of class $C^2$ (defined below) we show that the envelope of subsolutions is a locally $C^{1,1}$ solution (Steps 1–3). A stability estimate then gives a solution for continuous data (Step 4), and a comparison argument identifies every continuous solution with the envelope (Step 5).

The affine map $\tau (\xi )=a+R\xi $ carries the unit ball onto $B$. Since $|\tau (\xi )|^2-R^2|\xi |^2=2R\operatorname{Re}\sum _ j\xi _ j\bar a_ j+|a|^2$ is the real part of a holomorphic polynomial plus a constant, its complex Hessian vanishes, so $\tau $ pulls $\eta =\mathrm{dd}^{\mathrm{c}}|z|^2$ back to $R^2\eta $ and $\eta ^ n$ back to $R^{2n}\eta ^ n$. Since Bedford–Taylor products are invariant under biholomorphisms (proposition 7.2.2(5)), a function $u$ on $B$ satisfies $(\mathrm{dd}^{\mathrm{c}}u)^ n\ge f\eta ^ n$ if and only if $\xi \mapsto u(a+R\xi )$ satisfies the same inequality with the density $R^{2n}f(a+R\xi )$, whose $n$th root is $R^2f^{1/n}(a+R\xi )$; the same holds with equality. It therefore suffices to prove the theorem for the unit ball $B=B(0,1)$, with $R=1$ in (7.9.3). We write $z$ again for the variable and put

In this proof a subsolution for $(f,\varphi )$ is a bounded plurisubharmonic function $v$ on $B$ with $(\mathrm{dd}^{\mathrm{c}}v)^ n\ge f\eta ^ n$ and $\varlimsup _{B\ni z\to \zeta }v(z)\le \varphi (\zeta )$ for all $\zeta \in \partial B$. We say that the data $(f,\varphi )$ are of class $C^2$ if $F$ and $\varphi $ are restrictions of real $C^2$ functions on neighborhoods of $\overline B$ and $\partial B$, respectively.

Step 1. The envelope of subsolutions for data of class $C^2$. Assume that $(f,\varphi )$ is of class $C^2$. We first construct a subsolution with boundary values $\varphi $. Multiplying a $C^2$ extension of $\varphi $ by a smooth cutoff function for $\partial B$ in its domain which equals one near $\partial B$ (lemma 2.5.1(4)), we obtain a real $C^2$ function $b$ on $\mathbb C^ n$ with compact support and $b=\varphi $ on $\partial B$. Let $\lambda \ge 0$ bound the operator norm of the complex Hessian of $b$, and choose $c_0>\lambda $ with $(c_0-\lambda )^ n\ge \max _{\overline B}f$. The complex Hessian of $s=b+c_0\psi $ is at least $(c_0-\lambda )I$, so $s$ is plurisubharmonic (proposition 5.9.2) and $\det (s_{j\bar k})\ge (c_0-\lambda )^ n\ge f$ on $B$; by the formula for plurisubharmonic $C^2$ functions at the beginning of section 7.8, $(\mathrm{dd}^{\mathrm{c}}s)^ n\ge f\eta ^ n$ on $B$. Moreover $s$ is continuous on $\overline B$ and equals $\varphi $ on $\partial B$; in particular, $s$ is a subsolution. On the other hand, every subsolution is subharmonic (theorem 5.9.1) with boundary upper limits at most $\varphi $, so the comparison principle (theorem 3.4.5 and the remark following it) bounds it above by the harmonic extension $P\varphi $ of $\varphi $, its Poisson integral (theorem 3.1.18).

Let $H$ be the pointwise supremum of all subsolutions, so that $s\le H\le P\varphi $. By lemma 7.9.1, the maximum of finitely many subsolutions is a subsolution. By theorem 5.10.2, some countable family of subsolutions has the same upper semicontinuous regularization $H^*$ as the whole family, and its supremum equals $H^*$ almost everywhere. Taking finite maxima of $s$ and the members of this family, we obtain subsolutions $v_ j$ which increase to a function equal to $H^*$ almost everywhere, and $s\le v_ j\le P\varphi $. By theorem 7.6.1, $(\mathrm{dd}^{\mathrm{c}}v_ j)^ n\to (\mathrm{dd}^{\mathrm{c}}H^*)^ n$ weakly, and hence $(\mathrm{dd}^{\mathrm{c}}H^*)^ n\ge f\eta ^ n$. Since $P\varphi $ is continuous, $s\le H\le H^*\le P\varphi $; as $s$ and $P\varphi $ are continuous on $\overline B$ and equal $\varphi $ on $\partial B$, the function $H^*$ has boundary limits $\varphi $. Thus $H^*$ is a subsolution, so $H^*\le H$ and $H=H^*$. In particular, $H$ is a subsolution with $\lim _{B\ni z\to \zeta }H(z)=\varphi (\zeta )$ for all $\zeta \in \partial B$.

Step 2. Second differences for data of class $C^2$. We show that $H$ satisfies the hypothesis of lemma 7.7.2 on $B$. Fix a compact set $K\subset B$, and let $h_0>0$ and the automorphisms $S_{a,h}$, $a\in K$, $|h|\le h_0$, be as in lemma 7.7.1. Thus $(a,h,z)\mapsto S_{a,h}(z)$ is smooth, and holomorphic in $z$, on an open set $O$ containing the compact set $L=K\times \overline{B(0,h_0)}\times \overline B$, and so is the complex Jacobian determinant $J_{a,h}(z)=\det _{\mathbb C}DS_{a,h}(z)$ of $z\mapsto S_{a,h}(z)$. Since $S_{a,0}$ is the identity, $J_{a,0}=1$ on $B$, and hence on $\overline B$, as $z\mapsto J_{a,0}(z)$ is continuous. The open set $W=\{ (a,h,z)\in O:|J_{a,h}(z)-1|<1/2\} $ therefore contains the compact set $K\times \{ 0\} \times \overline B$, and hence all points within some positive distance of it; decreasing $h_0$, we may assume that $L\subset W$. Then $|J_{a,h}|^{2/n}=(J_{a,h}\overline{J_{a,h}})^{1/n}$ is smooth and positive on the neighborhood $W$ of $L$.

Since $S_{a,h}$ is holomorphic, $S_{a,h}^*(\mathrm dz_1\wedge \cdots \wedge \mathrm dz_ n)=J_{a,h}\, \mathrm dz_1\wedge \cdots \wedge \mathrm dz_ n$, and the conjugate identity holds for $\mathrm d\bar z_1\wedge \cdots \wedge \mathrm d\bar z_ n$. As $\eta ^ n$ is a constant multiple of $\mathrm dz_1\wedge \cdots \wedge \mathrm dz_ n\wedge \mathrm d\bar z_1\wedge \cdots \wedge \mathrm d\bar z_ n$, it follows that $S_{a,h}^*\eta ^ n=|J_{a,h}|^2\eta ^ n$. Pulling back the inequality $(\mathrm{dd}^{\mathrm{c}}H)^ n\ge F^ n\eta ^ n$ by $S_{a,h}$, and using again the invariance of Bedford–Taylor products under biholomorphisms, we obtain

Put

a nonnegative continuous function on $\overline B$. We claim that there are constants $C_1,C_2\ge 0$ such that, for all $a\in K$ and $|h|\le h_0$,

To prove the first inequality, fix $a\in K$ and $z\in \overline B$, and consider $\gamma (h)=F(S_{a,h}(z))|J_{a,h}(z)|^{2/n}$ as a function of the real variable $h\in \mathbb C^ n=\mathbb R^{2n}$, $|h|\le h_0$. Since $S_{a,h}(\overline B)=\overline B$, $F$ is of class $C^2$ on a neighborhood of $\overline B$, and the maps $(a,h,z)\mapsto S_{a,h}(z)$ and $(a,h,z)\mapsto |J_{a,h}(z)|^{2/n}$ are smooth near the compact set $L$, the second derivatives of $\gamma $ are bounded by a constant $2C_1$ which depends neither on $a$ nor on $z$. Taylor's formula with integral remainder, applied at $h$ and at $-h$, gives $|\gamma (h)+\gamma (-h)-2\gamma (0)|\le 2C_1|h|^2$, since the first-order terms cancel. As $\gamma (0)=F(z)$, it follows that $\Phi _{a,h}(z)=\frac12(\gamma (h)+\gamma (-h))\ge F(z)-C_1|h|^2$. The second inequality follows in the same way from a $C^2$ extension of $\varphi $ to a neighborhood of $\partial B$, since $S_{a,\pm h}$ map $\partial B$ onto itself and $S_{a,0}$ is the identity. The constants depend only on $K$, $h_0$, $n$ and the $C^2$ extensions of $F$ and $\varphi $.

Now consider

Since additive constants do not change Monge–Ampère measures, lemma 7.9.1, applied with $t=1/2$ and the quadratic term $C_1|h|^2|z|^2$, gives

Since $S_{a,\pm h}$ preserve $\partial B$ and $H$ has boundary limits $\varphi $, the boundary limits of $V_{a,h}$ are $\frac12(\varphi \circ S_{a,h}+\varphi \circ S_{a,-h})-C_2|h|^2\le \varphi $. Thus $V_{a,h}$ is a subsolution, and $V_{a,h}\le H$. Evaluating at $a$, where $S_{a,\pm h}(a)=a\pm h$, and using $-1\le \psi (a)\le 0$, we obtain

for all $a\in K$ and $|h|\le h_0$; here the points $a\pm h=S_{a,\pm h}(a)$ lie in $B$.

Step 3. The equation for data of class $C^2$. Since $K$ was arbitrary and $H$ is bounded and plurisubharmonic, lemma 7.7.2 applies on $B$. Thus $H$ is continuous and locally $C^{1,1}$ on $B$, and its distributional real Hessian is a locally bounded measurable matrix $A$. Almost every $x\in B$ satisfies

and the complex Hessian $(H_{j\bar k})$ computed from $A$ is positive semidefinite almost everywhere and satisfies $(\mathrm{dd}^{\mathrm{c}}H)^ n=n!\pi ^{-n}\det (H_{j\bar k})\, \mathrm dV_{2n}=\det (H_{j\bar k})\, \eta ^ n$. Since $(\mathrm{dd}^{\mathrm{c}}H)^ n\ge f\eta ^ n$, we have $\det (H_{j\bar k})\ge f$ almost everywhere, and it remains to prove the reverse inequality.

Suppose, to the contrary, that $\det (H_{j\bar k}(x))>f(x)$ at a point $x$ at which the Taylor expansion holds and $(H_{j\bar k}(x))$ is positive semidefinite. The quadratic Taylor polynomial $Q(x+h)=H(x)+\nabla H(x)\cdot h+\frac12h^ tA(x)h$ has the constant complex Hessian $(H_{j\bar k}(x))$, which is positive semidefinite with positive determinant, hence positive definite. Choose $\varepsilon >0$ so small that the matrix $(H_{j\bar k}(x))-\varepsilon I$ is positive definite and its determinant exceeds $f(x)$. By the continuity of $f$ and the Taylor expansion at $x$, there is $r>0$ such that $\overline{B(x,r)}\subset B$, this determinant exceeds $f$ on $B(x,r)$, and $|H-Q|<\varepsilon r^2/2$ on $\partial B(x,r)$. The function

is plurisubharmonic (proposition 5.9.2), and $(\mathrm{dd}^{\mathrm{c}}q)^ n=\det \bigl ((H_{j\bar k}(x))-\varepsilon I\bigr )\eta ^ n\ge f\eta ^ n$ on $B(x,r)$. Moreover $q<H-\varepsilon r^2/4$ on $\partial B(x,r)$, while $q(x)=H(x)+\varepsilon r^2/4>H(x)$.

Let $\widehat H=\max \{ H,q\} $ on $B(x,r)$ and $\widehat H=H$ on $B\setminus B(x,r)$. Since $q$ and $H$ are continuous and $q<H$ on the sphere $\partial B(x,r)$, we have $q<H$ on a neighborhood of this sphere. Hence $\widehat H$ is plurisubharmonic by proposition 5.9.5, and it coincides with $H$ near $\partial B(x,r)$. On $B(x,r)$, lemma 7.9.1, applied with $F=G=f^{1/n}$, gives $(\mathrm{dd}^{\mathrm{c}}\widehat H)^ n\ge f\eta ^ n$. Every other point of $B$ has a neighborhood on which $\widehat H=H$, so $(\mathrm{dd}^{\mathrm{c}}\widehat H)^ n\ge f\eta ^ n$ near it as well. Since every nonnegative test function on $B$ is a finite sum of nonnegative test functions supported in such neighborhoods (lemma 2.5.2), the inequality holds on all of $B$. Thus $\widehat H$ is a subsolution with $\widehat H(x)>H(x)$, a contradiction. Consequently $\det (H_{j\bar k})=f$ almost everywhere, that is, $(\mathrm{dd}^{\mathrm{c}}H)^ n=f\eta ^ n$. As $H$ is continuous on $B$ with boundary limits $\varphi $ (Step 1), it extends to a solution in $\mathrm{PSH}(B)\cap C(\overline B)$, which is locally $C^{1,1}$ on $B$.

Step 4. Stability and continuous data. Let $(f,\varphi )$ and $(g,\chi )$ be data of class $C^2$, with envelopes $H_{f,\varphi }$ and $H_{g,\chi }$ as in Step 1. Put $\delta =\| \varphi -\chi \| _{L^\infty (\partial B)}$ and $\sigma =\| f^{1/n}-g^{1/n}\| _{L^\infty (B)}$. By lemma 7.9.1,

and $H_{g,\chi }+\sigma \psi -\delta $ has boundary limits $\chi -\delta \le \varphi $. It is therefore a subsolution for $(f,\varphi )$, and since $\psi \ge -1$ we get $H_{g,\chi }-\sigma -\delta \le H_{f,\varphi }$. Exchanging the roles of the two data gives (7.9.3) with $R=1$ for data of class $C^2$.

Now let $f$ and $\varphi $ be continuous. Extend $F$ to $\mathbb C^ n$ by $F(z)=F(z/|z|)$ for $|z|>1$ and convolve with mollifiers; this gives smooth functions $F_ j\ge 0$ with $F_ j\to F$ uniformly on $\overline B$ (lemma 2.5.1(1)). Similarly, extending $\varphi $ to a neighborhood of $\partial B$ by $\varphi (z)=\varphi (z/|z|)$ and mollifying, we obtain smooth functions $\varphi _ j$ on a neighborhood of $\partial B$ with $\varphi _ j\to \varphi $ uniformly on $\partial B$. The data $(F_ j^ n,\varphi _ j)$ are of class $C^2$, and by Steps 1–3 the corresponding envelopes $H_ j=H_{F_ j^ n,\varphi _ j}$ are solutions which are continuous on $\overline B$ with boundary values $\varphi _ j$. By the estimate just proved,

so $(H_ j)$ is a uniform Cauchy sequence on $\overline B$. Its uniform limit $U$ is continuous on $\overline B$ and equal to $\varphi $ on $\partial B$. By proposition 7.2.6, $U$ is plurisubharmonic on $B$ and $(\mathrm{dd}^{\mathrm{c}}H_ j)^ n\to (\mathrm{dd}^{\mathrm{c}}U)^ n$ weakly; since $F_ j^ n\to f$ uniformly on $\overline B$, we obtain $(\mathrm{dd}^{\mathrm{c}}U)^ n=f\eta ^ n$.

Step 5. The envelope, uniqueness and stability. Let $U$ be a solution of the problem that is continuous on $\overline B$, for instance the one constructed in Step 4, and let $v$ be a subsolution for $(f,\varphi )$. For $\varepsilon >0$ put $v_\varepsilon =v+\varepsilon \psi $. By lemma 7.9.1, $(\mathrm{dd}^{\mathrm{c}}v_\varepsilon )^ n\ge (F+\varepsilon )^ n\eta ^ n$, while $(\mathrm{dd}^{\mathrm{c}}U)^ n=F^ n\eta ^ n$. Since $\varliminf _{z\to \zeta }(U(z)-v_\varepsilon (z))\ge 0$ for every $\zeta \in \partial B$, the comparison principle theorem 7.3.11 gives

These integrals are finite, because $F$ is bounded on $B$, and $(F+\varepsilon )^ n-F^ n\ge \varepsilon ^ n$, so the Borel set $\{ U<v_\varepsilon \} $ is Lebesgue null. Hence $\max \{ U,v_\varepsilon \} =U$ almost everywhere, and the uniqueness assertion of theorem 5.9.1 gives $U\ge v_\varepsilon $ everywhere. Letting $\varepsilon \downarrow 0$, we get $U\ge v$. Since $U$ is itself a subsolution, it is the pointwise supremum of all subsolutions. In particular, any two solutions in $\mathrm{PSH}(B)\cap C(\overline B)$ coincide with this supremum, which proves uniqueness. For data of class $C^2$ the solution is therefore the envelope of Step 1, which is locally $C^{1,1}$ by Step 3.

Finally, if $(f,\varphi )$ and $(g,\chi )$ are continuous data, approximate both as in Step 4. By uniqueness, the corresponding solutions converge uniformly to $H_{f,\varphi }$ and $H_{g,\chi }$, and (7.9.3) for the approximating data of class $C^2$ passes to the limit. Undoing the affine change of variables proves all assertions of the theorem. $\square$