Strictly plurisubharmonic function

Definition

A $C^2$ function $\phi$ on $U\subseteq\mathbb C^d$ is strictly plurisubharmonic if its Levi form is positive definite at every point. Quantitatively, one often assumes

\mathcal L_\phi(z;v)\ge\kappa(z)|v|^2

for a positive function $\kappa$ .

Nonsmooth convention

For an upper-semicontinuous function, strict plurisubharmonicity may be defined locally by requiring $\phi-\varepsilon|z|^2$ to be plurisubharmonic for some $\varepsilon>0$ . Distributional inequalities permit measurable lower bounds $\kappa$ .

Role in the d-bar equation

The positive lower bound supplies coercivity in the Hörmander $L^2$ $\bar\partial$ -existence theorem, controlling a solution by the data divided by $\kappa$ .

References

Lars Hörmander, An Introduction to Complex Analysis in Several Variables, 3rd ed., North-Holland, 1990.