, Volume 240, Issue 4, pp 823-834

Solving the Gleason problem on linearly convex domains

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Abstract.

Let \(\Omega\) be a bounded, connected linearly convex set in \({\mathbf{C}} ^n\) with \(C^{1+\epsilon}\) boundary. We show that the maximal ideal (both in \(H^{\infty}(\Omega)\) ) and \(A^{m}(\Omega), 0 \leq m \leq \infty\) ) consisting of all functions vanishing at \(p \in \Omega\) is generated by the coordinate functions \(z_1 - p_1, \ldots, z_n - p_n\) .

Received: 2 July 2001; in final form: 26 September 2001 / Published online: 28 February 2002