Theorem 11.34.11F4FDB7BC8A03

Proof. The continuous trace of $\mathrm{dd}^{\mathrm{c}}u$ is the Euclidean Laplacian of $u$, up to the fixed normalization. Local constant-coefficient elliptic regularity gives $u\in W^{2,p}$ for every finite $p$. Indeed, subtract the Newtonian potential of a cutoff of this trace. Its second high-frequency derivatives are order-zero multipliers, and the low-frequency kernels are locally bounded; the remaining distribution is harmonic. Sobolev embedding gives a $C^{1,\alpha }$ representative for every $\alpha <1$, which agrees with the quasi-psh representative.

On a coordinate ball choose smooth $\rho $ with $\mathrm{dd}^{\mathrm{c}}\rho \geq \omega $. The radial regularizations of the bounded psh function $u+\rho $ decrease to it, and their complex Hessians converge uniformly on smaller balls. Continuity of bounded products and the finite smooth-background expansion identify $(\omega +\mathrm{dd}^{\mathrm{c}}u)^ n$ with $\sigma ^ n$. For smooth real $h$, the same argument applies to $u+th$, which is admissible for all sufficiently small signed $t$ by uniform positivity. Define

Its equation has the same normalization scalar as the equation for $u$, so $B_0(F_ t)=B_0(F)$. The supporting inequality gives $\int _ X(F_ t-F)\, \mathrm d\nu \leq 0$ for each $\nu \in \mathcal A_0(F)$. Since $F_ t-F=tL_\sigma h+O(t^2)$ uniformly, using both signs yields stationarity.

Choose $p=4n+1$ and $q=p/(p-1)=1+1/(4n)$. By corollary 12.9.3, every supporting probability has an $L^ q$ density, since $q<n/(n-1)$. Smooth density in $W^{2,p}$ extends its stationarity to every $W^{2,p}$ test. By lemma 11.24.1, all these densities lie in a one-dimensional space. Their integrals equal one, so they coincide. Nonemptiness is theorem 11.34.2, and the derivative formula follows from proposition 11.34.5. $\square$