Obstructions to uniform estimates for solutions to the d-bar equation

Abstract

We show that if, for every bounded d-bar-closed (0,1)-form f, a pseudoconvex domain \Omega admits a solution to $\bar\partial u=f$ that is continuous up to the boundary and has uniform estimates in terms of $\|f\|_\infty$, then each p\in bdy(\Omega) must necessarily admit a peak function in the class $A(\Omega):=\mathcal{O}(\Omega)\cap\mathcal{C}(\overline\OM)$. We use this fact to examine some geometrical obstructions to uniform estimates for the d-bar equation.