Sharp pointwise and uniform estimates for $\bar\partial$

TL;DR

This paper derives sharp pointwise and uniform estimates for solutions to the $ar ext{d}$-problem on convex and classical domains using weighted $L^2$ methods, revealing new blow-up behaviors.

Contribution

It introduces novel sharp pointwise estimates for the $ar ext{d}$-problem on specific domains, including Cartan classical domains, and characterizes blow-up orders.

Findings

Pointwise estimates are sharp and demonstrate specific blow-up rates.

Maximum blow-up order exceeds previous bounds in certain domains.

Uniform estimates are established under stronger conditions on $f$.

Abstract

We use weighted $L^2$-methods to obtain sharp pointwise estimates for the canonical solution to the equation $\bar\partial u=f$ on smoothly bounded strictly convex domains and the Cartan classical domain domains when $f$ is bounded in the Bergman metric $g$. We provide examples to show our pointwise estimates are sharp. In particular, we show that on the Cartan classical domains $\Omega$ of rank $2$ the maximum blow up order is greater than $-\log \delta_\Omega(z)$, which was obtained for the unit ball case by Berndtsson. For example, for IV$(n)$ with $n \geq 3$, the maximum blow up order is $\delta(z)^{1 -{n \over 2}}$ because of the contribution of the Bergman kernel. Additionally, we obtain uniform estimates for the canonical solutions on the polydiscs, strictly pseudoconvex domains and the Cartan classical domains under stronger conditions on $f$.