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

Abstract

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.