1
$\begingroup$

The openness theorem says that:

If $\varphi$ be a negative plurisubharmonic function in the unit ball $B(0,1)$ in $\mathbb{C}^{n}$ satisfying $$ \intop_{B(0,1)}e^{-\varphi}<\infty, $$ then there exist $p>1$ and $r>0$ such that $$ \intop_{B(0,r)}e^{-p\varphi}<\infty. $$

My question is: is the above theorem trivial if we don't assume $\varphi\left(0\right)=-\infty$? That is, if, say, $\varphi\left(0\right)=-1$ then it's trivial to prove the above theorem.

$\endgroup$
0
$\begingroup$

It's known that if $h$ is PSH near $0$ and $e^{-h}$ is not locally integrable at $0$ then $h(0)=-\infty$, see e.g. Degenerate Complex Monge-Ampere Equations, Vincent Guedj and Ahmed Zeriahi. This is not a trivial result, however in the context of $\varphi\left(0\right)=-\infty$ in your question, it's trivial.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.