Estimates for the \({\bar{\partial }}\) -Equation on Canonical Surfaces

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.