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

Phillip S. Harrington; Andrew Raich

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arXiv · 1512.02523

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

Phillip S. Harrington·Andrew Raich

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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.

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BibTeX RIS

Phillip S. Harrington, Andrew Raich. 2015-12-08. A remark on boundary estimates on unbounded $Z(q)$ domains in $\mathbb{C}^n$. https://arxiv.org/abs/1512.02523

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