On Gleason's decomposition for \(A^{\infty}(\bar D)\) (Q1077596)

Let D be a bounded pseudoconvex domain in \({\mathbb{C}}^ n\) with \(C^{\infty}\)-boundary. Let \(A^{\infty}(\bar D)\) be the algebra of holomorphic functions in D having derivatives of all orders continuous up to the boundary. The author proves that for each \(f\in A^{\infty}(\bar D)\) there is a decomposition \(f(z)-f(w)=\sum^{n}_{i=1}g_ i(z,w)(z_ i-w_ i)\) with \(g_ i\in A^{\infty}(\bar D\times \bar D)\). The result is in general not true for an arbitrary domain in \({\mathbb{C}}^ n\).