Estimates for the ∂¯ -Equation on Canonical Surfaces

We study the solvability in Lp 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.