Global analytic hypoellipticity of the\bar \partial-Neumann problem on circular domains

In this paper we show that if \(D \subseteq \mathbb{C}^n ,n \geqq 2\) , is a smooth bounded pseudoconvex circular domain with real analytic defining functionr(z) such that \(\sum\limits_{k = 1}^n {z_k \frac{{\partial r}}{{\partial z_k }}} \ne 0\) for allz near the boundary, then the solutionu to the \(\bar \partial\) -Neumann problem, $$square u = (\bar \partial \bar \partial * + \bar \partial *\bar \partial )u = f,$$ is real analytic up to the boundary, if the given formf is real analytic up to the boundary. In particular, if \(D \subseteq \mathbb{C}^n ,n \geqq 2\) , is a smooth bounded complete Reinhardt pseudoconvex domain with real analytic boundary. Then ? is analytic hypoelliptic.