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Corollary 5.14.2079C564A098F9

Corollary 5.14.20.79C564A098F9 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

\[ \varlimsup _{\Omega '\ni y\to x}\psi (y)\leq \varphi (x) \]

for any $x\in \Omega \cap \partial \Omega '$. Define

\[ \eta (z)=\begin{cases} \varphi (z)\lor \psi (z),& \quad \textup{if }z\in \Omega ';\\ \varphi (z),& \quad \textup{if }z\in \Omega \setminus \Omega '. \end{cases} \]

Then $\eta \in \mathrm{PSH}(\Omega )$.

Proof. Let $u$ be the function equal to $\max\{\varphi,\psi\}$ on $\Omega'$ and to $\varphi$ elsewhere. The boundary limsup hypothesis is exactly the gluing condition in the maximum gluing proposition. Hence $u$ is plurisubharmonic. This argument is local near each point of $\partial\Omega'\cap\Omega$; no relative compactness of $\Omega'$ in $\Omega$ is required. Away from that boundary, the assertion is simply stability under the maximum or the original plurisubharmonicity of $\varphi$.

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