Failure of global regularity of (partial derivative)over-bar(b) on a convex domain with only one flat point

In this paper we exhibit a bounded domain in C-2 with real analytic boundar y which is strictly convex except at one point and for which the (partial-d erivative)over-bar(b) operator is not analytic hypoelliptic modulo its kern el. The importance of such an example is twofold: First it shows that the theor em of Boas and Straube on global C-infinity regularity for (partial-derivat ive)over-bar(b) on convex domains cannot be extended to the analytic case; secondly it is the first example of nonanalytic hypoellipticity of (partial -derivative)over-bar(b) on a domain with isolated weakly pseudoconvex point s in the boundary.