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 type m.

The equation $\bar \partial u = f$ is solved in Ω with sharp estimates: if f has bounded coefficients, the coefficients of our solution u are in the Lipschitz space Λ. Optimal estimates are also given when data have coefficients belonging to L^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 of bΩ, we introduce a weight expressed in terms of the Bergman kernel of Ω.