Precise subelliptic estimates for a class of special domains

TL;DR

This paper improves subelliptic estimates for the $ar ext{-} ext{Neumann}$ problem on certain domains, providing sharper bounds and a simplified proof using Catlin's method of weight functions.

Contribution

It establishes better subelliptic estimates for regular coordinate domains, surpassing previous bounds, and simplifies the proof process.

Findings

Improved $ar ext{-} ext{Neumann}$ estimates with better $ ext{epsilon}$ bounds.

Simplified proof technique based on weight functions and Levi form estimates.

Applicable to specific classes of domains with regular coordinates.

Abstract

For the $\bar\partial$-Neumann problem on a regular coordinate domain $\Omega\subset \C^{n+1}$, we prove $\epsilon$-subelliptic estimates for an index $\epsilon$ which is in some cases better than $\epsilon=\frac1{2m}$ ($m$ being the {\it multiplicity}) as it was previously proved by Catlin and Cho in \cite{CC08}. This also supplies a much simplified proof of the existing literature. Our approach is founded on the method by Catlin in \cite{C87} which consists in constructing a family of weights $\{\phi^\delta\}$ whose Levi form is bigger than $\delta^{-2\epsilon}$ on the $\delta$-strip around $\partial\Omega$.