A remark on boundary estimates on unbounded $Z(q)$ domains in $\mathbb{C}^n$

TL;DR

This paper investigates the connection between the Folland-Kohn basic estimate and the $Z(q)$-condition on unbounded domains in complex space, establishing equivalences and implications related to pseudoconvexity and boundary estimates.

Contribution

It proves that on unbounded pseudoconvex domains, the Folland-Kohn estimate is equivalent to uniform strict pseudoconvexity, and on non-pseudoconvex domains, it links the estimate to the $Z(q)$-condition.

Findings

Folland-Kohn estimate is equivalent to uniform strict pseudoconvexity on unbounded pseudoconvex domains.

The estimate fails on the Siegel upper half space despite its strict pseudoconvexity.

On non-pseudoconvex domains, the estimate implies a uniform $Z(q)$ condition.

Abstract

The goal of this note is to explore the relationship between the Folland-Kohn basic estimate and the $Z(q)$-condition. In particular, on unbounded pseudoconvex (resp., pseudoconcave) domains, we prove that the Folland-Kohn basic estimate is equivalent to uniform strict pseudoconvexity (resp., pseudoconcavity). As a corollary, we observe that despite the Siegel upper half space being strictly pseudoconvex and biholomorphic to a the unit ball, it fails to satisfy uniform strict pseudoconvexity and hence the Folland-Kohn basic estimate fails. On unbounded non-pseudoconvex domains, we show that the Folland-Kohn basic estimate on $(0,q)$-forms implies a uniform $Z(q)$ condition, and conversely, that a uniform $Z(q)$ condition with some additional hypotheses implies the Folland-Kohn basic estimate for $(0,q)$-forms.