Failure of global regularity of \(\overline {\partial}_b\) on a convex domain with only one flat point (Q1306238)

By constructing a concrete example of a bounded domain in \(\mathbb{C}^{2}\), the author shows the failure of global regularity of the \(\overline{\partial_{b}}\)-operator on a convex domain with only one flat point. More precisely, there exists a bounded domain \(\Omega\) in \(\mathbb{C}^{2}\) with real analytic boundary \(M\), which is pseudoconvex, of finite type, and strictly pseudoconvex except at one point where the \(\overline{\partial_{b}}\)-operator is not globally analytic hypoelliptic modulo its kernel. This result asserts that Boas's and Straube's theorem on global regularity for \(\overline{\partial_{b}}\) on convex domain cannot be extended to the analytic case, and it is also the first example of non analytic hypoellipticity of \(\overline{\partial_{b}}\) on a domain with isolated weakly pseudoconvex points in the boundary.