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

TL;DR

The paper investigates geometric obstructions to uniform estimates for solutions to the d-bar equation in pseudoconvex domains, showing that boundary peak functions are necessary for such estimates to hold.

Contribution

It establishes a link between the existence of boundary peak functions and the possibility of uniform estimates for the d-bar equation solutions.

Findings

Boundary peak functions are necessary for uniform estimates.

Certain geometric obstructions prevent uniform estimates.

The paper characterizes when uniform estimates can or cannot hold.

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.