Sharp Estimates for the $\bar{\partial}$-Neumann Problem on Regular Coordinate Domains

TL;DR

This paper establishes sharp algebraic lower bounds for subelliptic estimates in the $ar{ ext{d}}$-Neumann problem on regular coordinate domains, linking geometric properties to algebraic multiplicities.

Contribution

It provides a novel algebraic lower bound for the subelliptic gain in the $ar{ ext{d}}$-Neumann problem on regular coordinate domains, connecting boundary geometry with algebraic multiplicity.

Findings

Largest subelliptic gain is bounded below by the inverse of twice the ideal's multiplicity.

The bound is purely algebraic and depends on boundary point properties.

Results improve understanding of regular coordinate domain estimates.

Abstract

This paper treats subelliptic estimates for the $\bar{\partial}$-Neumann problem on a class of domains known as regular coordinate domains. Our main result is that the largest subelliptic gain for a regular coordinate domain is bounded below by a purely algebraic number, the inverse of twice the multiplicity of the ideal associated to a given boundary point.