Algorithms for subelliptic multipliers in $\mathbb{C}^2$

TL;DR

This paper investigates the limitations of the Kohn algorithm for subelliptic estimates in certain pseudoconvex domains in ^2 and proposes modifications to achieve effective subellipticity proofs, with extensions to higher dimensions.

Contribution

It identifies failures of the Kohn algorithm in specific domains and introduces modified algorithms to establish subellipticity effectively, extending results to higher dimensions.

Findings

Kohn algorithm fails to provide effective bounds in some finite type domains

Modified algorithms can prove subellipticity in ^2 domains with real analytic boundary

Results are extended to higher-dimensional complex spaces

Abstract

We give examples of pseudoconvex domains of finite type in $\mathbb{C}^2$ where the Kohn algorithm for subelliptic estimates fails to yield an effective lower bound for the order of subellipticity in terms of the type. We show how to modify the algorithm to obtain an effective procedure to prove subellipticity on domains of finite type in $\mathbb{C}^2$ with real analytic boundary satisfying a condition slightly stronger than pseudoconvexity. We close with a generalization to higher dimensions.