Regularity up to the boundary for the \(\overline\partial\) complex (Q1962711)

The author shows that, under a suitable weak-pseudoconvexity condition at a point \(z_0\) of the boundary of a wedge \(W\) in \(\mathbb C^n\), there exists a fundamental system of open neighborhoods \(\{U\}\) of \(z_0\) such that the \(\mathcal C^\infty(\overline{W\cap U})\) Dolbeault cohomology groups vanish in degree \(j>k\). The condition is formulated in terms of two positive integers \(a\), \(q\) and \(k=\max\{a,q\}\). The wedges considered are intersections of \(m\) half-spaces with \(\mathbb C\)-linearly independent exterior normals at \(z_0\). Results on the vanishing of the local cohomology on wedges had also been obtained before by many authors, see for instance \textit{R. A. Ajrapetyan} and \textit{G. M. Henkin}, Russ. Math. Serv. 39, No. 3, 41-118 (1984); translation from Usp. Mat. Nauk 39, No. 3(237), 39-106 (1984; Zbl 0589.32035) and \textit{M. Nacinovich}, Math. Ann. 268, 449-471 (1984; Zbl 0574.32045).