Optimal Lipschitz Estimates for the $\bar{\partial }$ equation on a class of convex domains

Abstract

In this paper, we consider the Cauchy-Riemann equation $\bar{\partial }u=f$ in a new class of convex domains in $\C^n.$ We prove that under $L^p$ data, we can choose a solution in the Lipschitz space ${\Lambda }_{\alpha },$ where $\alpha$ is an optimal positive number given explicitly in terms of $p.$