Local real analytic boundary regularity of the \(\partial\)-equation on convex domains (Q1203580)

Using a well-known integral solution operator of the \(\overline\partial\)- equation obtained by Henkin, the author proves the following theorem. Theorem. Let \(\Omega\) be a bounded convex domain in \(\mathbb{C}^ n\) with \(C^ 2\) boundary and \(\xi\) be a boundary point of \(\Omega\). If \(\partial\Omega\) is totally convex at \(\xi\) in the complex tangential directions, then for any \(\overline\partial\)-closed \(f\in C^ 1_{(p,q+1)}(\overline\Omega)\) \((p,q>0)\) which is real analytic at \(\xi\), there exists a solution \(u\) of the \(\overline\partial\)-equation \(\overline\partial u=f\) in \(\Omega\) such that \(u\) is also real analytic at \(\xi\).