In the late ten years, the resolution of the equation∂ u = f with sharp estimates has been intensively studied for convex domains of finite type in C n by many authors. Generally they used kernels constructed with holomorphic support function satisfying "good" global estimates. In this paper, we consider the case of lineally convex domains. Unfortunately, the method used to obtain global estimates for the support function cannot be carried out in that case. Then we use a kernel that does not gives directly a solution of the∂ -equation but only a representation formula which allows us to end the resolution of the equation using Kohn's L 2 theory.As an application we give the characterization of the zero sets of the functions of the Nevanlinna class for lineally convex domains of finite type.The general scheme of the proof is identical to the one used in the convex case and consists in three steps. First, for the general case of geometrically separated domains, we prove some "Malliavin conditions" on closed positive (1, 1)-currents Θ and then we solve the equation dw = Θ with good estimates. The third step, which solves the∂ -equation for (0, 1)-form with L 1 estimates on the boundary, is only done in the case of lineally convex domains of finite type: Theorem 1.2. Let Ω be a bounded lineally convex domain of finite type in C n with smooth boundary. Let f be a (0, 1)-form in Ω whose coefficients are C 1 (Ω) functions and which is∂ -closed. Then there exists a solution of the equation ∂ u = f , smooth on Ω and continuous on Ω such that u L 1 (∂ Ω) ≤ C||| f ||| k (see Section 2.1 formula (2.3)), the constant C depending only on Ω. In other words, there exists a solution of the equation ∂ b u = f , in the sense of [Sko76], in L 1 (∂ Ω).
SOLUTIONS FOR THE∂ -EQUATION FOR LINEALLY CONVEX DOMAINS OF FINITE TYPEFirst of all, we recall the definition of lineally convex domain: Definition 2.1. A domain Ω in C n , with smooth boundary is said to be lineally convex at a point p ∈ ∂ Ω if there exists a neighborhood W of p such that, for all point z ∈ ∂ Ω ∩W ,where T 10 z is the holomorphic tangent space to ∂ Ω at the point z. Furthermore, we always suppose that ∂ Ω is of finite type at every point of ∂ Ω ∩ W . Shrinking W if necessary, we may assume that there exists a C ∞ defining function ρ for Ω and a number η 0 > 0 such that ∇ρ(z) = 0 at every point of W and the level sets {z ∈ W such that ρ(z) = η}, −η 0 ≤ η ≤ η 0 , are lineally convex of finite type.As we want to obtain global results, we need these properties at every boundary point. Thus, in all our work, by "lineally convex domain" we mean a bounded smooth domain having a (global) defining function satisfying the previous hypothesis at every point of ∂ Ω.In Section 2.1 we define a punctual anisotropic norm for forms, . k , related to the geometry of the domain (formula (2.2)). With this notation, the main goal of this Section is to prove the following reformulation of Theorem 1.2 for (0, q)-forms:2000 Mathematics Subject Classification. 32F17, 32T25, 32T40.
