Sharp pointwise and uniform estimates for $\bar\partial$

Abstract

We use weighted L 2 -methods initiated by Berndtsson to obtain sharp pointwise estimates for the canonical solution to the equation ∂u = f on bounded convex homogeneous domains and on smoothly bounded strictly convex domains Ω in C n when |f | g is bounded in Ω with the Bergman type metric g. Additionally, we obtain uniform estimates for the canonical solutions on polydiscs, strictly pseudoconvex domains and Cartan classical domains under stronger conditions than the boundedness of |f | g . We provide examples to show our pointwise estimates are sharp. In particular, we show that if Ω is a Cartan classical domain of rank 2 then the maximum blow up order is greater than that of the unit ball, − log δ Ω (z), which was obtained by Berndtsson. For example, for IV(n) with n ≥ 3, the maximum blow up order is δ(z) 1− n 2 because of the contribution from the Bergman kernel.