Suppose D is a relatively compact pseudoconvex domain in \({\mathbb{C}}^ n\) with \(C^ 1\) boundary. Suppose the \({\bar \partial}\)-operator on D is hypoelliptic in the sense that for every \({\bar \partial}\)-closed (0,1)- form \(\alpha\) on D with coefficients in \(L^ 2\) we can find an \(L^ 2\) function f with \({\bar \partial}f=\alpha\) and whose singular support is contained within the singular support of \(\alpha\). The author proves that, in this case, every local CR-function on the boundary \(\partial D\) extends locally into D as a holomorphic function.
When D has \(C^ 2\) boundary this result follows from results of Catlin, Diederich-Pflug, and Trépreau but the author’s method, in addition to allowing a rougher boundary, is much more direct. He also discusses related results on analytic convexity.