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Let Ω be a smoothly bounded convex domain of finite type in C n . We show that a divisor in Ω satisfying the Blaschke condition (respectively associated to a current of order a > 0) can be defined by a function in the Nevanlinna class N 0 (Ω) (respectively the Nevanlinna-Djrbachian class N a (Ω)). The proof is based on L 1 (bΩ) estimates (resp. weighted L 1 (Ω) estimates) for the solution of the∂-equation on Ω.

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Let Q be a bounded convex domain in C n, with smooth boundary of finite type m.The equation c)u~f is solved in f~ with sharp estimates: if f has bounded coefficients, the coefficients of our solution u are in the Lipschitz space A1/m(f~). OptimM estimates are Mso given when data have coefficients belonging to LP([~), p> 1.We solve the c)-equation by means of integral operators whose kernels are not based on the choice of a "good" support function. Weighted kernels are used; in order to reflect the geometry of bf~, we introduce a weight expressed in terms of the Bergman kernel of f/.