# A characterization of plurisubharmonic functions

Let $$\Omega\subset \mathbb{C}^n$$ be an open subset. Let $$u\colon \Omega\to [-\infty,+\infty)$$ be an upper semi-continuous function.

Recall that $$u$$ is called plurisubharmonic (psh) if its restriction to any complex line is subharmonic.

Any psh function $$u$$ satisfies the following property: for any point $$x\in \Omega$$ and for any $$C^2$$-smooth function $$\phi$$ defined near $$x$$ and such that $$u\leq \phi$$ and $$u(x)=\phi(x)$$ one has $$(\Delta_L (\phi|_L))(x)\geq 0$$ for any complex line $$L$$ containing the point $$x$$. Here $$\Delta_L$$ denotes the Laplacian on the line $$L$$.

Is the converse true, i.e. if an upper semi-continuous function $$u$$ satisfies the above condition is it psh? A reference would be very helpful.

This post is a continuation of A possible characterization of subharmonic functions

## 2 Answers

You can consult Harvey and Lawson, sections 5 and 6, on that matter. Especially Lemma 5.5 and point (6) on p. 19 (note that for smooth $$\phi$$ condition you gave is equivalent to having complex hessian non negative).

• Do you mean that the answer to my question is positive?
– asv
Commented Jan 19, 2023 at 7:24
• Yes, their "original" definition of psh there is "geometric" i.e. subharmonic on complex lines and they argue that using their Lemma 5.5 gives it being equivalent to viscosity psh in the sens: viscosity contestant have to have complex hessian positive. For smooth functions this is clearly equivalent to the condition you want so you have the same viscosity condition. Commented Jan 19, 2023 at 15:27

Answers to this and the preceding question can be obtained from various definitions/characterizations of harmonicity and plurisubharmonicity to be found in the book

Klimek, Maciej Pluripotential theory. London Mathematical Society Monographs. New Series, 6. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1991. xiv+266 pp. ISBN: 0-19-853568-6

Of particular interest to you should be Theorem 2.4.1 (several characterizations of subharmonicity), Theorem 2.5.1 (subharmonicity and Laplacian), Theorem 2.9.1 (a characterization of plurisubharmonicity) and Theorem 2.9.11 (plurisubharmonicity and distributional derivatives).

• Could you please add more details how the statements you mentioned imply the result I need. Thank you.
– asv
Commented Jan 4, 2023 at 6:27