Proposition 5.15.6 (Bedford–Taylor)80A580DBD009

Proof. By 5.15.5, there is an open neighborhood $U\subseteq \Omega $ of $x$ and $\psi \in \mathrm{PSH}(U)$ such that

Without loss of generality we may assume $x=0$ and that $U=B_ n(0,1)$ is the open unit ball.

Since $\psi $ is upper semicontinuous at $0$ with $\psi (0)=0$, we may choose a strictly decreasing sequence $(\delta _ j)_{j\geq 1}\subseteq (0,\mathrm{e}^{-1})$ such that

In particular $|\log \delta _ j|>1$ for every $j\geq 1$, so the series of constants

converges. We further set

which is also decreasing in $j$.

For each $j\geq 1$ define

On the boundary $|z|=\delta _ j$, the logarithmic term tends to $-2^{-j-1}$. By (5.15.7),

for every $z_0$ with $|z_0|=\delta _ j$. Hence the boundary condition in 5.14.16 is satisfied. We obtain that $\varphi _ j\in \mathrm{PSH}(\mathbb {C}^ n)$.

By construction, $\varphi _ j\leq 0$ on $U$, $\varphi _ j(0)=-2^{-j}$, and for $z\in E\setminus \{ 0\} $ with $|z|<\gamma _ j$ we have $\varphi _ j(z)\leq -1$, since

and, on $E\setminus \{ 0\} $, $\psi (z)-2^{-j}=-1-2^{-j}<-1$.

We then define

On the unit ball $U$ each $\varphi _ j\leq 0$, so the partial sums $S_ N\coloneqq \sum _{j=1}^ N\varphi _ j$ form a decreasing sequence of plurisubharmonic functions with

so $\varphi |_ U\in \mathrm{PSH}(U)$ by proposition 5.14.2. On $\mathbb {C}^ n\setminus B_ n(0,\delta _1)$ each $\varphi _ j$ equals $\frac{2^{-j-1}}{|\log \delta _ j|}\log |z|$, and (5.15.8) shows $S_ N$ converges uniformly on every compact subset to $\bigl (\sum _{j\geq 1}\tfrac {2^{-j-1}}{|\log \delta _ j|}\bigr )\log |z|$, a plurisubharmonic limit. Both formulas agree on $U\setminus B_ n(0,\delta _1)$, so altogether $\varphi \in \mathrm{PSH}(\mathbb {C}^ n)$.

We now verify the three claims.

Property 2. This is the equality $\varphi (0)=-\sum _{j\geq 1}2^{-j}=-1>-\infty $ obtained above.

Property 3. Fix $j_0\geq 1$. For $z\in E\setminus \{ 0\} $ with $|z|<\gamma _{j_0}$, the bound $\varphi _ j(z)\leq -1$ holds for every $j\leq j_0$ (since the $\gamma _ j$ are decreasing), while $\varphi _ j(z)\leq 0$ for $j>j_0$. Hence $\varphi (z)\leq -j_0$, and letting $j_0\to \infty $ yields

Property 1. Fix any $z_0\in \mathbb {C}^ n$ with $z_0\neq 0$ and choose $J\geq 1$ such that $\delta _ J<|z_0|/2$. On the open set $W=\{ w\in \mathbb {C}^ n:|w|>|z_0|/2\} $, all $\varphi _ j$ with $j\geq J$ lie in the outer regime, and (5.15.8) shows $\sum _{j\geq J}\varphi _ j(w) =\bigl (\sum _{j\geq J}\tfrac {2^{-j-1}}{|\log \delta _ j|}\bigr )\log |w|$ is locally bounded on $W$. The remaining finitely many terms $\varphi _1,\ldots ,\varphi _{J-1}$ are plurisubharmonic on $\mathbb {C}^ n$, hence locally bounded above on $W$, while $\varphi _ j(w)\geq \tfrac {2^{-j-1}}{|\log \delta _ j|}\log |w|$ keeps them locally bounded below as well. Therefore $\varphi $ is locally bounded on $W$. Since $z_0\neq 0$ was arbitrary, $\varphi $ is locally bounded on $\mathbb {C}^ n\setminus \{ 0\} $, which is property 1. $\square$