ar X iv : m at h / 01 06 23 5 v 1 [ m at h . C V ] 2 7 Ju n 20 01 SOLVING THE GLEASON PROBLEM ON LINEARLY CONVEX DOMAINS

Let Ω be a bounded domain in C. Let R(Ω) (usually A(Ω) or H(Ω)) be a ring of holomorphic functions that contains the polynomials, and let p = (p1, . . . , pn) be a point in Ω. Recall the Gleason problem, cf. [8]: is the maximal ideal in R(Ω) consisting of functions vanishing at p, generated by the coordinate functions (z1 − p1), . . . , (zn − pn) ? One says that a domain Ω has the Gleason R-property if this is the case for all points p ∈ Ω. We also say that it has the Gleason-property with respect to R(Ω). Leibenzon was the first to solve a non trivial Gleason problem. He proved ([11]) that the Gleason problem can be solved on any convex domain in C having a Cboundary. This result was sharpened by Grangé ([9], for H(Ω)), and by Backlund and Fällström ([3] and [4], for H(Ω) and A(Ω) respectively), for convex domains in C having only a C-boundary. Using his theorem on solvability of the ∂-problem ([14]), Øvrelid proved in [15] that a strictly pseudoconvex domain in C with C-boundary has the Gleason Aproperty. Fornæss and Øvrelid showed in [7] that a pseudoconvex domain in C with real analytic boundary has the Gleason A-property. This was extended by Noell ([13]) to pseudoconvex domains in C having a boundary of finite type. Backlund and Fällström proved in [6] that a bounded, pseudoconvex Reinhardt domain in C with C-boundary that contains the origin, has the Gleason A-property. The present authors showed in [12] that one does not need that Ω is pseudoconvex or that it contains the origin. They also solved the H-problem for such Reinhardt domains. Note that there are not always solutions to the Gleason problem; in fact, Backlund and Fällström showed ([5]) that there even exists an H-domain of holomorphy on which the problem is not solvable. In this article, we return to the original method of Leibenzon, and use it to solve the Gleason problem on C-convex domains (these are domains such that their intersection with any complex line passing through the domain is connected and simply connected) in C with C-boundary. We denote the derivate of a function g with