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
Best Answer
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).