Theorem 5.9.12A4F77263F48

Proof. Let $u\in \mathrm{PSH}(\Omega )$. Being upper semicontinuous, $u$ is bounded above on compact sets. Fix $x\in \Omega $ and $r>0$ with $\overline{B(x,r)}\subset \Omega $, and let $M$ be an upper bound for $u$ on $\overline{B(x,r)}$. For every unit vector $a\in \mathbb {C}^ n$ the closed disc $\{ x+ta:|t|\leq r\} $ lies in $\Omega $, and the restriction $t\mapsto u(x+ta)$ is subharmonic, possibly identically $-\infty $, on a neighborhood of this disc. The monotonicity of circle means in theorem 3.4.5 therefore gives

With $\tau $ as in (5.8.3), we integrate this inequality over $a\in S^{2n-1}$ with respect to $\tau $; by Tonelli's theorem, applied to the nonnegative function $M-u$, and by the invariance of $\tau $ under $a\mapsto \mathrm{e}^{\mathrm{i}\theta }a$, the right-hand side becomes the sphere mean $\int _{S^{2n-1}}u(x+ra)\, \mathrm{d}\tau (a)$. Thus $u$ satisfies the submean inequality of definition 3.4.2 at every point, so $u$ is subharmonic. Since $\Omega $ is connected and $u\not\equiv -\infty $, proposition 3.4.8 shows that $u\in L^1_{\mathrm{loc}}(\Omega )$. Since, moreover, the restriction of $u$ to each connected component of each affine complex-line section of $\Omega $ is subharmonic, possibly identically $-\infty $, lemma 5.8.2 gives $\mathrm{dd}^{\mathrm{c}}u\ge 0$.

Conversely, let $T$ be a real distribution with $\mathrm{dd}^{\mathrm{c}}T\geq 0$. Then $\Delta T=4\sum _ jT_{j\bar j}\geq 0$, so by theorem 3.4.12 $T$ has a canonical subharmonic representative $v$, which is locally integrable and satisfies (3.4.13). Since $\mathrm{dd}^{\mathrm{c}}v=\mathrm{dd}^{\mathrm{c}}T\geq 0$, proposition 5.4.4 shows that the restriction of $v$ to each connected component of each affine complex-line section of $\Omega $ is subharmonic or identically $-\infty $. As $v$ is upper semicontinuous and locally integrable, hence not identically $-\infty $, it is plurisubharmonic. If $w$ is another plurisubharmonic representative of $T$, then $w$ is a locally integrable subharmonic function by the first part, and $w=v$ almost everywhere; by the uniqueness in theorem 3.4.12, $w=v$ everywhere. $\square$