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

Abstract

We study the solvability in L p of the∂-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∂ s and∂ w of∂. For∂ s we have solvability, whereas for∂ 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∂ introduced by the first and fourth author, and we study mapping properties of these operators at the singularity.