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In this paper we exhibit a bounded domain in C 2 with real analytic boundary which is strictly convex except at one point and for which the @b operator is not analytic hypoelliptic modulo its kernel. The importance of such an example is twofold: First it shows that the theorem of Boas and Straube on global C 1 regularity for @b on convex domains cannot be extended to the analytic case; secondly it is the rst example of non analytic hypoellipticity of @b on a domain with isolated weakly pseudoconvex points in the boundary.

Failure of global regularity of ∂b on a convex domain with only one flat point