Uniform estimates for the canonical solution to the $\bar\partial$-equation on product domains

TL;DR

This work proves a uniform $L^{\infty}$-estimate for the canonical solution to $\bar{\partial}u=f$ on product domains $\Omega=\prod_j D_j$ with $C^2$ planar boundaries, for $f$ continuous up to $\bar{\Omega}$. It develops a product-domain canonical solution by replacing slice Cauchy operators with one-dimensional canonical kernels, first for differentiable data and then for continuous data via a derivative-free integral formula and an exhaustion/stability argument. The main result shows $\|u\|_{\infty} \le C\|f\|_{\infty}$ with $C$ depending only on $\Omega$, and ensures $u$ is continuous on $\Omega$, thereby answering an open Kerzman question and extending Landucci’s bidisc result to higher dimensions. The approach leverages Green's function bounds for planar domains, stability under domain exhaustion, and a detailed kernel decomposition to control derivatives, with additional consequences for the Bergman projection and related $L^p$ estimates.

Abstract

We obtain uniform estimates for the canonical solution to $\bar\partial u=f$ on the Cartesian product of bounded planar domains with $C^2$ boundaries, when $f$ is continuous up to the boundary. This generalizes Landucci's result for the bidisc toward higher dimensional product domains. In particular, it answers an open question of Kerzman for continuous datum.

Theorems & Definitions (31)

• Theorem 1.1
• Lemma 2.1: Kerzman Ker76
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Corollary 2.4
• Lemma 3.1
• Proposition 3.2
• proof