\(\mathcal C^k\)-estimates for the Cauchy-Riemann operator on domains with \(q\)-convex and \(q\)-concave edges
In this highly technical paper, \({\mathcal C}^k\) estimates for the solutions of the \(\overline\partial\) CR equation are established for \(q\)-convex edges. This sharpens previous estimates by \textit{M.-Y. Barkatou} [Math. Z. 239, No. 2, 335--352 (2002; Zbl 1006.32031)]. Then analogous estimates for \(q\)-concave edges are provided. From these results, a result for local resolution with regularity up to the boundary, on the complement \(W\) of a \(q\)-convex domain \(D\) with \({\mathcal C}^d\) piecewise boundaries is given. This provides some improvement of earlier results by Laurent-Thiebault andLeiterer. Finally, the author establishes a global resolution result for \((0, r)\)-forms with compact support on the complement of a strictly \(q\)-convex domain \(\Omega\Subset X\) with piecewise smooth boundaries. This gives a generalization to the separation theorem by Andreotti and Vesentini.
scientific article; zbMATH DE number 1965495