Hörmander L2 theorem for the d-bar equation

Statement

Let $\phi:\mathbb C^d\to\mathbb R$ be plurisubharmonic with Levi form bounded below in the distributional sense by $\kappa(z)I$ , where $\kappa>0$ . If a ((0,1))-form $\eta$ satisfies $\bar\partial\eta=0$ and

\int_{\mathbb C^d}|\eta|^2\frac{e^{-\phi}}{\kappa}<\infty,

then there is a distributional solution $g$ of $\bar\partial g=\eta$ such that

\int_{\mathbb C^d}|g|^2e^{-\phi} \le \int_{\mathbb C^d}|\eta|^2\frac{e^{-\phi}}{\kappa}.

Geometric input

The operator $\bar\partial$ is the Dolbeault operator from Dolbeault theory. Positivity of the Levi form is the curvature term that makes the weighted estimate coercive.

Holomorphic correction

If $h$ is a smooth cutoff and $g$ solves $\bar\partial g=\bar\partial h$ , then $f=h-g$ is entire. Choosing a weight with a strong singular or convex term near a point can make $g$ small there, ensuring $f$ is nonzero.

References

Lars Hörmander, “ $L^2$ estimates and existence theorems for the $\bar\partial$ operator,” Acta Mathematica 113 (1965), 89–152. DOI record.
Lars Hörmander, An Introduction to Complex Analysis in Several Variables, 3rd ed., North-Holland, 1990.