Sobolev regularity for the $\overline\partial$-Neumann operator and transverse vector fields

TL;DR

On bounded smooth pseudoconvex domains in $\mathbb{C}^n$ with $n>2$, the paper introduces a new sufficient condition for the exact Sobolev regularity of the $\bar{\partial}$-Neumann operator $N_q$ via a family of transverse $(1,0)$-vector fields satisfying a strong variant of property ($\widetilde{P}$), denoted ($\widetilde{P}^{\#}$). Employing Harrington's vector-field framework and elliptic regularization, the authors derive precise commutator controls for $[D_{X_{\alpha,\varepsilon}}^k,\bar{\partial}]$ and $[D_{X_{\alpha,\varepsilon}}^k,\bar{\partial}^*]$, establish weighted $L^2$ estimates, and prove that $N_q$ is continuous on $W^{k}_{0,q'}(\Omega)$ for all $k\ge0$ and $q'\ge q$; in particular, they obtain global regularity for $N_{n-1}$ and extend to all $N_q$ via downward induction. The approach also shows biholomorphic invariance of the property and recovers Boas–Straube-type regularity in the presence of real good vector fields. Collectively, the results provide a robust geometric-analytic pathway to exact regularity beyond strictly or strongly pseudoconvex domains, connecting to Zhang's compactness-type criteria and the Diederich–Fornæss index.

Abstract

On a bounded smooth pseudoconvex domain in $\mathbb{C}^n$ with $n >2$, inspired by the compactness condition introduced by Yue Zhang, we present the new sufficient condition for the exact regularity of the $\overline\partial$-Neumann operator via the transverse vector fields.

Theorems & Definitions (41)

• Theorem 1.1
• Theorem 1.2
• Corollary 2.1
• Theorem 2.1
• Lemma 2.1
• proof
• Proposition 2.1
• proof
• Lemma 2.2
• proof