Sharp estimates for $\bar \partial $ on convex domains of finite typeon convex domains of finite type

Abstract

Let Ω be a bounded convex domain in C n , 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 toL p(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.