Supnorm estimates for $\bar{\partial }$ on product domains in ${\mathbb{C}}^n$

Abstract

Let $\Omega \subset {\mathbb{C}}^n$ be a product of one-dimensional open bounded domains with $C^{1,\alpha }$ boundary, where $0<\alpha <1$. Using methods from complex analysis in one variable, we construct an integral operator that solves $\bar{\partial }$ in $\Omega$ with supnorm estimates when the datum is in $C^{n-1,\alpha }(\Omega )$.

v3: similar to v1. Theorem 6.4 of v2 has been removed, since the proof was incomplete. 20 pages

Cited by in corpus (1)

• Optimal Hölder regularity for the $\bar{\partial }$ problem on product domains in ${\mathbb{C}}^2$