Lemma 5.14.16.49CC56CB484C Let $\Omega \subseteq \mathbb {C}^ n$ be a domain and $\Omega '\subseteq \Omega $ be a subdomain. Consider $\varphi \in \mathrm{PSH}(\Omega )$ and $\psi \in \mathrm{PSH}(\Omega ')$. Assume that
504311AA3518
5.14.17
\begin{equation} \varliminf _{\substack {\Omega '\ni y\to x,\\ \psi (y)\neq -\infty }}\bigl (\varphi (y)-\psi (y)\bigr )\geq 0 \end{equation}
for any $x\in \Omega \cap \partial \Omega '$. Define
\[ \eta (z)=\left\{ \begin{aligned} \varphi (z)\lor \psi (z),& \quad \text{if }z\in \Omega ',\\ \varphi (z),& \quad \text{if }z\in \Omega \setminus \Omega '. \end{aligned} \right. \]
Then $\eta \in \mathrm{PSH}(\Omega )$.
Proof.
See 5.14.18 for the notations used in the proof.
Take $\epsilon >0$. We first define
\[ \eta _{\epsilon }(z)=\begin{cases} \varphi (z)\lor (\psi (z)-2\epsilon ),& \textup{if }z\in \Omega ';\\ \varphi (z),& \textup{if }z\in \Omega \setminus \Omega '. \end{cases} \]
We claim that
\[ \eta _{\epsilon }\in \mathrm{PSH}(\Omega ). \]
By our assumption, for each $x\in \Omega \cap \partial \Omega '$, we can find an open neighborhood $U_ x\subseteq \Omega $ such that for any $y\in U_ x\cap \Omega '$, we have $\varphi (y)\geq \psi (y)-\epsilon $. Therefore, by taking a union of the $U_ x$'s for various $x$, there is an open neighborhood $U$ of $\Omega \cap \partial \Omega '$ such that
\[ \varphi (y)\geq \psi (y)-\epsilon ,\quad \forall y\in U\cap \Omega '. \]
Since $U$ contains $\Omega \cap \partial \Omega '$, the set $(\Omega \setminus \Omega ')\cup U$ is open in $\Omega $. On this open set we have $\eta _{\epsilon }=\varphi $, hence $\eta _{\epsilon }$ is plurisubharmonic there. It is plurisubharmonic on $\Omega '$ by proposition 5.14.2. These two descriptions agree on the overlap $U\cap \Omega '$, so our claim follows from the local nature of plurisubharmonicity.
Next we observe that as $\epsilon $ decreases to $0$, the functions $\eta _{\epsilon }$ increase to $\eta $. The family $(\eta _\epsilon )_{0<\epsilon \leq 1}$ is locally uniformly bounded from above. This is clear away from $\Omega \cap \partial \Omega '$. Near $\Omega \cap \partial \Omega '$, applying the previous estimate with $\epsilon =1$ gives an open neighborhood $U_1$ of $\Omega \cap \partial \Omega '$ such that $\psi \leq \varphi +1$ on $U_1\cap \Omega '$, hence $\eta _\epsilon \leq \varphi +1$ on $U_1$ for all $0<\epsilon \leq 1$. Therefore, $\eta ^*\in \mathrm{PSH}(\Omega )$ by proposition 5.14.2.
It remains to observe that $\eta $ is upper semicontinuous. This is only non-trivial on the boundary of $\Omega '$. Take $x\in \Omega \cap \partial \Omega '$ and let $(y_ i)_{i>0}$ be a sequence in $\Omega '$ with limit $x$. We first show that
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5.14.19
\begin{equation} \varlimsup _{i\to \infty } \psi (y_ i)\leq \varphi (x). \end{equation}
We may assume that $\psi (y_ i)\neq -\infty $ for all $i>0$ and the left-hand side of (5.14.19) is not $-\infty $. Then we can compute
\[ \varlimsup _{i\to \infty } \psi (y_ i)\leq \varlimsup _{i\to \infty } \psi (y_ i)+\varliminf _{i\to \infty } \bigl (\varphi (y_ i)-\psi (y_ i)\bigr )\leq \varlimsup _{i\to \infty }\varphi (y_ i)\leq \varphi (x). \]
Combining this with the upper semicontinuity of $\varphi $, we get
\[ \varlimsup _{i\to \infty }\bigl (\varphi (y_ i)\lor \psi (y_ i)\bigr )\leq \varphi (x)=\eta (x). \]
This proves upper semicontinuity across $\Omega \cap \partial \Omega '$. Therefore, $\eta =\eta ^*\in \mathrm{PSH}(\Omega )$.
$\square$
Pome
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