Local solvability for \(\overline{\partial}\) with Gevrey regularity up to the boundary of a \(r\)- convex domain
Let \(\Omega\) be an open set in \(\mathbb{C}^ n\) of class \(C^ 3\) and \(\zeta\) a point of its boundary \(b\Omega\). \(\Omega\) is \(q\)-convex near \(\zeta\) if its Levi forms has at least \(r\) nonnegative eigenvalues near \(\zeta\) in \(b \Omega\). In this situation, the solvability of \(\overline \partial\), with \(C^ \infty\) regularity up to the boundary, is unknown. However, in this work, we show that if \(a \in [1,3/2[\), \(\overline \partial\) is solvable near \(\zeta\) in \(\overline \Omega\), with Gevrey regularity of class \(a\), for \(\overline \partial\)-closed \((p,q)\)-forms of Gevrey regularity of class \(a\) if \(q \geq n - r\). The proof of this result relies essentially on like Hörmander \(L^ 2\) techniques and on a method used by Dufresnoy to solve \(\overline \partial\) on certain closed sets in \(\mathbb{C}^ n\).
scientific article; zbMATH DE number 737189
scientific article