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

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 $(\ast )$ 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.

21 pages

References in corpus (2)

• Koppelman formulas on affine cones over smooth projective complete intersections
• Koppelman formulas on the A_1-singularity