An approximation theorem for $\overline \partial$-closed forms of type $(n, n-1)$

Abstract:

Let $D$ be a bounded open set in ${C^n}$ with smooth boundary. Then every closed form of type $(n,n - 1)$ which is ${C^\infty }$ on $\bar D$ can be approximated uniformly on $\bar D$ by $(n,n - 1)$ forms which are closed in a neighborhood of $\bar D$. If ${C^n} - D$ is connected these forms can be chosen to be closed in ${C^n}$. This is applied to prove that a continuous function on the connected boundary of a bounded domain in ${C^n}$ admits a holomorphic extension to the interior if and only if it is a weak solution of the tangential Cauchy-Riemann equations.