Koppelman formulas and the \(\dbar\)-equation on an analytic space

Abstract

Let \(X\) be an analytic space of pure dimension. We introduce a formalism to generate intrinsic weighted Koppelman formulas on \(X\) that provide solutions to the \(\dbar\)-equation. We prove that if \(\phi\) is a smooth \((0,q+1)\)-form on a Stein space \(X\) with \(\dbar\phi=0\), then there is a smooth \((0,q)\)-form \(\psi\) on \(X_{reg}\) with at most polynomial growth at \(X_{sing}\) such that \(\dbar\psi=\phi\). The integral formulas also give other new existence results for the \(\dbar\)-equation and Hartogs theorems, as well as new proofs of various known results.