Estimates for the $\bar{\partial}$-equation on canonical surfaces

TL;DR

This paper investigates the solvability of the $ar{ ext{d}}$-equation in $L^p$ spaces near canonical surface singularities, providing a detailed analysis for the case $p=2$ and exploring different extensions of the operator.

Contribution

It offers a comprehensive analysis of $ar{ ext{d}}$-equation solvability near du Val singularities, including conditions for different operator extensions and the use of integral operators.

Findings

$ar{ ext{d}}_s$ extension is solvable near singularities.

$ar{ ext{d}}_w$ extension solvability depends on boundary condition $(*)$.

Mapping properties of integral operators are crucial for the analysis.

Abstract

We study the solvability in $L^p$ of the $\bar\partial$-equation in a neighborhood of a canonical singularity on a complex surface, a so-called du Val singularity. We get a quite complete picture in case $p=2$ for two natural closed extensions $\bar\partial_s$ and $\bar\partial_w$ of $\bar\partial$. For $\bar\partial_s$ we have solvability, whereas for $\bar\partial_w$ there is solvability if and only if a certain boundary condition $(*)$ is fulfilled at the singularity. Our main tool is certain integral operators for solving $\bar\partial$ introduced by the first and fourth author, and we study mapping properties of these operators at the singularity.