Local $L^2$ results for $\overline{\partial}$ on a singular surface

Abstract

The Cauchy-Riemann equations are fundamental in complex analysis. This paper contributes to the understanding of these equations on singular spaces. Various methods have been used to overcome the problem of defining forms near singularities. One can blow up the singularity, restrict forms from smooth ambient spaces or work on the regular points. In this paper we use the latter approach to obtain square integrable solutions on singular surfaces. This can be briefly called the Kohn solution up to the singularity to contrast with results in terms of curvature, weights or different function spaces.