The Kohn Algorithm on Denjoy-Carleman Classes

TL;DR

This paper extends the equivalence between finite type conditions and subellipticity of the $ar ext{ ext}{}$-Neumann problem to domains with defining functions in Denjoy-Carleman quasianalytic classes, involving advanced algebraic geometry techniques.

Contribution

It introduces a novel approach using algebraic geometry over non-Noetherian Denjoy-Carleman rings to analyze the $ar ext{ ext}{}$-Neumann problem in complex analysis.

Findings

Established equivalence in Denjoy-Carleman classes

Proved the Denjoy-Carleman ring satisfies the $ oot{acc}$ property

Extended finite type and subellipticity results to new function classes

Abstract

The equivalence of the Kohn finite ideal type and the D'Angelo finite type with the subellipticity of the $\bar\partial$-Neumann problem is extended to pseudoconvex domains in $C^n$ whose defining function is in a Denjoy-Carleman quasianalytic class closed under differentiation. The proof involves algebraic geometry over a ring of germs of Denjoy-Carleman quasianalytic functions that is not known to be Noetherian and that is intermediate between the ring of germs of real-analytic functions and the ring of germs of smooth functions. It is also shown that this type of ring of germs of Denjoy-Carleman functions satisfies the $\sqrt{acc}$ property, one of the strongest properties a non-Noetherian ring could possess.