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I came across the following exercise on plurisubharmonic functions (Krantz, function theory of several complex variables, §111):

Let $\Omega \subseteq \mathbb{C}^n$ be a domain, $f: \Omega \to \mathbb{R}$ continuous. Suppose that for each compact $K \subset \subset \Omega$ and for each $h \in C(K)$ that is pluriharmonic function on $ \mathring { K } $, it holds that $h\ge f$ on $ \partial K$ implies $h \ge f$ on $K$. Then $f$ is plurisubharmonic.

Is it true or false? Any help or insight is deeply appreciated.

Nikodem Dyzma
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unicornki
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One of the many equivalent definitions of plurisubharmonicity is being subharmonic on every complex line, i.e. $f:\Omega \rightarrow \mathbb{R}$ is plurisubharmonic if the function $\zeta \rightarrow f(a+b\zeta)$, restricted to $\{\zeta \in \mathbb{C}\; | \; a +b\zeta \in \Omega\}$ is subharmonic, for any $a,b \in \mathbb{C}^n$. The same way we can define pluriharmonic functions, i.e being harmonic on every complex line.

So want to show now that $f$ is plurisubharmonic. Take a complex line $l$ that intersects $\Omega$ and take some point $z \in \Omega \cap l$. Since subharmonicity is a local property, if we show that there exists a neighbourhood of $z$ in $\Omega \cap l$, on which $f$ is subharmonic, we are done. Take any small ball $B(z,r)$ that is relatively compat in $\Omega$ and any pluriharmonic $h$ that domninates $f$ on $\partial B(z,r)$. Then $\Omega \cap l \cap B(z,r)$ is again a ball on this line, $h$ is harmonic and dominates $f$ on the boundary. From assumption that means that $h\ge f$ on the whole $\Omega \cap l \cap B(z,r)$ and that implies subharmonicity of $f$ on this line.

Nikodem Dyzma
  • 343