Sharp estimates for
$$\bar \partial $$
 on convex domains of finite typeon convex domains of finite type
AbstractLet Ω be a bounded convex domain in Cn, with smooth boundary of finite typem.The equation
$$\bar \partial u = f$$
 is solved in Ω with sharp estimates: iff has bounded coefficients, the coefficients of our solutionu are in the Lipschitz space Λ. Optimal estimates are also given when data have coefficients belonging toLp(Ω),p≥1.We solve the
$$\bar \partial $$
-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 ofbΩ, we introduce a weight expressed in terms of the Bergman kernel of Ω.