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The Kohn Algorithm on Denjoy-Carleman Classes

Andreea C. Nicoara

TL;DR

This work extends Kohn's three-way equivalence between subellipticity of the $\bar{\partial}$-Neumann problem, Kohn finite ideal type, and finite D'Angelo type from the real-analytic setting to domains with boundaries in Denjoy–Carleman quasianalytic classes closed under differentiation. It develops an algebraic-geometry framework over non-Noetherian germ rings $C_M$ (with $C_M$, $C^{\mathbb R}_M$) that nonetheless enjoy $\sqrt{acc}$ and Łojasiewicz inequalities via Bierstone–Milman resolution, enabling a Nullstellensatz and coherent-sheaf techniques. A Denjoy–Carleman version of the Kohn algorithm is introduced, with a modified chain $\tilde{I}^q_k$ in $C_{M,b\Omega}$, and it is shown that $I^q_k = \widehat{\tilde{I}^q_k}$, preserving quasicoherence and allowing the use of resolution of singularities. The central result proves that termination of the Kohn algorithm (and hence subellipticity) is equivalent to finite D'Angelo type in this setting, arguing by contrapositive that non-termination would imply a boundary holomorphic variety of dimension at least $q$, which contradicts finite type; the framework potentially extends to broader quasianalytic/definable classes, offering a path toward a Kohn conjecture in non-Noetherian contexts.

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.

The Kohn Algorithm on Denjoy-Carleman Classes

TL;DR

This work extends Kohn's three-way equivalence between subellipticity of the -Neumann problem, Kohn finite ideal type, and finite D'Angelo type from the real-analytic setting to domains with boundaries in Denjoy–Carleman quasianalytic classes closed under differentiation. It develops an algebraic-geometry framework over non-Noetherian germ rings (with , ) that nonetheless enjoy and Łojasiewicz inequalities via Bierstone–Milman resolution, enabling a Nullstellensatz and coherent-sheaf techniques. A Denjoy–Carleman version of the Kohn algorithm is introduced, with a modified chain in , and it is shown that , preserving quasicoherence and allowing the use of resolution of singularities. The central result proves that termination of the Kohn algorithm (and hence subellipticity) is equivalent to finite D'Angelo type in this setting, arguing by contrapositive that non-termination would imply a boundary holomorphic variety of dimension at least , which contradicts finite type; the framework potentially extends to broader quasianalytic/definable classes, offering a path toward a Kohn conjecture in non-Noetherian contexts.

Abstract

The equivalence of the Kohn finite ideal type and the D'Angelo finite type with the subellipticity of the -Neumann problem is extended to pseudoconvex domains in 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 property, one of the strongest properties a non-Noetherian ring could possess.

Paper Structure

This paper contains 4 sections, 20 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. The Denjoy-Carleman Quasianalytic Classes
  3. Subellipticity of the $\bar{\partial}$-Neumann Problem and the Kohn Algorithm
  4. Equivalence of Types

Key Result

Theorem 1.2

Let $x_0$ be a point in ${\mathbb R}^n$ or ${\mathbb C}^n,$ and let $C_M (x_0)$ be a ring of germs of Denjoy-Carleman quasianalytic functions closed under differentiation. The local ring $C_M (x_0)$ has the $\sqrt{acc}$ property. In other words, if $\mathcal{I}_1 \subset \mathcal{I}_2 \subset \cdots

Theorems & Definitions (33)

  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5
  • Proposition 2.7
  • Proposition 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 23 more