10.31.37 Ordinary types and bounded carriers11CC2B98E362

Proof. The proof has four steps: two inputs of minimal singularity type, two inputs of the ordinary type of $\varphi $, one input of that type, and finally an input with a bounded carrier measure.

It suffices to prove the result for $\lambda =1$. Indeed, replace $(\theta ,\varphi ,u,v)$ by $(\lambda \theta ,\lambda \varphi ,\lambda u,\lambda v)$: envelopes and relative classes scale with the potentials, volumes and Monge–Ampère measures scale by $\lambda ^ n$, and a bounded-carrier domination remains one after changing its constant. Dividing the resulting solution by $\lambda $ gives the desired equation. Similarly, a common downward translation of $u$, $v$ and the solution allows us to assume $u,v\leq \varphi \leq 0$; by the envelope identity, these inequalities hold after a sufficiently large common translation. We restore the translation at the end.

Step 1: inputs of minimal singularity type. First take $\varphi =V$ and $u,v\sim V$. Truncating only the weights, define the finite nonzero measures

The functions $\max (u,-j)$ need not be $\theta $-plurisubharmonic; they occur only as weights. Theorem 10.31.22 gives $a_ j\in \mathcal E_ V$ with $\mu _{a_ j}=\mathrm{e}^{a_ j}\tau _ j$.

Choose $D\geq \tfrac 12\| u-v\| _\infty +n\log 2$, where the difference is taken off the common poles, and put $w=(u+v)/2-D$. Positivity of mixed products gives

Exponential order in $\mathcal E_ V$ (corollary 13.19.3) gives $a_ j\geq w$. Since $\tau _ j$ increases with $j$, the same order shows that $a_ j$ decreases. The envelope identity gives $a_1\leq V+\sup _ Xa_1$, so $a_1\sim V$ by the lower bound $w$. Consequently

The decreasing limit $a\geq w$ is not identically $-\infty $ and has the ordinary type of $V$. Fixed-dominator continuity (theorem 10.31.5) gives weak convergence of the left-hand sides, and the right-hand sides converge in total variation by dominated convergence with respect to $\mu _ u+\mu _ v$. Therefore

Comparing this equation with each input measure, exponential order now gives $a\leq u,v$. This comparison uses the limiting equation; the truncated weights alone would not give it.

Step 2: two inputs of the ordinary type of $\varphi $. Suppose $u,v\sim \varphi $, and put $u_ j=\max (u,V-j)$ and $v_ j=\max (v,V-j)$. Step 1 gives solutions $a_ j\sim V$ with

The differences $u_ j-v_ j$ satisfy the same uniform bound as $u-v$. With $D$ as above, $w_ j=(u_ j+v_ j)/2-D$ satisfies the reverse inequality for this equation, and exponential order gives

The fixed lower bound $w=(u+v)/2-D$ has the ordinary type of $\varphi $, and hence lies in $\mathcal E_\varphi $. Extract an $L^1$ limit $a\not\equiv -\infty $; it satisfies $w\leq a\leq \min (u,v)$, so $a\sim \varphi $. The input masses are at most $M_ V$. Lemma 10.31.35, with $\sigma =V$ and lower bound $h=w$, proves convergence in capacity and the equation. The lemma applies although the approximating solutions have the type of $V$ and the limit has the type of $\varphi $.

Step 3: one input of the ordinary type of $\varphi $. Now let $u\in \mathcal E_\varphi $ and $v\sim \varphi $, and set $u_ j=\max (u,\varphi -j)$. Step 2 gives $a_ j\sim \varphi $ with

We construct a common lower bound in $\mathcal E_\varphi $. Put $\delta =n\log 2>0$ and

By stretching and contact, $s_ j\leq a_ j$ and

On $D_ j=\{ s_ j\leq u-2\delta \} $, contact implies $a_ j\leq u_ j-\delta $. Since $\mathrm{e}^{-\delta }=2^{-n}$, the $\mu _{u_ j}$ terms cancel in (10.31.39). Also $v\geq \varphi -C\geq u_ j-C$ for a fixed constant $C$. Hence

All measures being subtracted here are finite.

For $N\geq 1$ form the finite rooftops

They belong to $\mathcal E_\varphi $ by the finite-rooftop proposition. On the set $\{ R_ N\leq u-2\delta \} $, partition the contact set according to the first index $i$ with $R_ N=s_ i$. Since $u_ i\geq u_ N$, (10.31.40) gives

The contact pieces are disjoint, so there is no factor $N$. On $\{ q_ N\leq u-2\delta \} $ only the contact part $q_ N=R_ N$ contributes. Since $u\leq u_ N$ and $t \mathrm{e}^{-t/2}\leq 2/\mathrm{e}$ for $t\geq 0$, we obtain

The first term bounds the contribution of the complementary set, including any contact with $u$. The first-moment criterion proposition 10.31.8, with fixed reference $u$, implies that the decreasing sequence $q_ N$ has a limit $h\in \mathcal E_\varphi $, not identically $-\infty $. For each fixed $j$ and every $N\geq j$, we have $q_ N\leq s_ j\leq a_ j$, so $h\leq a_ j$ for every $j$.

The $a_ j$ now have a subsequence converging in $L^1$ to a limit not identically $-\infty $, bounded above by $u$ and $v$. Lemma 10.31.35, with reference $\varphi $, fixed second input $v$ and common lower bound $h$, proves the required equation in $\mathcal E_\varphi $; here the input masses are at most $M$.

Step 4: a bounded carrier measure for the second input. Finally let $u,v\in \mathcal E_\varphi $ and $\mu _ v\leq A\nu $, with $\nu =\mathrm{MA}_\omega (\psi )$ and $\psi $ bounded. Again put $u_ j=\max (u,\varphi -j)$. By Step 3, applied with the inputs interchanged, there are $a_ j\in \mathcal E_\varphi $ satisfying the same equations and bounds as in Step 3.

Let $r=P_\theta (u,v)\in \mathcal E_\varphi $. On $\{ a_ j<r\} $ one has $r\leq u_ j,v$, so the inequality $t \mathrm{e}^{-t}\leq 1/\mathrm{e}$ gives

Since $a_ j\leq \varphi $, the one-sided first-moment criterion gives a uniform lower bound for $\sup _ Xa_ j$. By lemma 5.44.10,

Applying this to $a_ j-\sup _ Xa_ j$ and using the lower bound for the suprema, we obtain

Keep $s_ j=Q_2(a_ j,u_ j)$, $C_ j$ and $D_ j$ from Step 3. The same cancellation in (10.31.39), now using only $a_ j\leq v$, gives

Take one additional rooftop,

On its contact set with $u-2\delta $ its measure is at most $\mu _ u$; on its other contact part, where $q_ j=s_ j\leq u-2\delta $, the preceding inequality applies. Thus

Moreover the integrand $u-2\delta -q_ j$ vanishes on the first contact part. On the other part, the measure is carried by $C_ j$, where $u_ j-s_ j=2(u_ j-a_ j)$. Therefore

which is uniformly bounded by (10.31.41). The first-moment criterion with reference $u-2\delta $ prevents collapse of the $q_ j$. Extract $L^1$ limits of the $a_ j$ and of the $q_ j$, not identically $-\infty $. The fixed-dominator theorem applies to (10.31.42) and, after a further extraction, gives

Since $q_ j\leq s_ j\leq a_ j$, this is a common lower bound for the extracted solutions, and lemma 10.31.35 gives the equation. Its truncation levels are the original indices of the retained subsequence.

In every step the solution lies below $u$ and $v$, so its measure is bounded by $\mu _ u+\mu _ v$. Two solutions have the same exponentially weighted measure, so corollary 13.19.3, applied in both directions, gives uniqueness, even when that weighted measure has infinite mass. Restoring the common translation and the exponent finishes the proof. $\square$