Direct proof of termination of the Kohn algorithm in the real-analytic case

In 1979 J.J. Kohn gave an indirect argument via the Diederich–Fornæss Theorem showing that finite D’Angelo type implies termination of the Kohn algorithm for a pseudoconvex domain with real-analytic boundary. We give here a direct argument for this same implication using the stratification coming from Catlin’s notion of a boundary system as well as algebraic geometry on the ring of real-analytic functions. We also indicate how this argument could be used in order to compute an effective lower bound for the subelliptic gain in the $\overline{\partial}$-Neumann problem in terms of the D’Angelo type, the dimension of the space, and the level of forms provided that an effective Łojasiewicz inequality can be proven in the real-analytic case and slightly more information obtained about the behavior of the sheaves of multipliers in the Kohn algorithm.

Keywords

pseudoconvex domains, Kohn algorithm, Catlin multitype, finite D’Angelo type, Nullstellensatz, Łojasiewicz inequality, quasi-flasque sheaf

2010 Mathematics Subject Classification

Primary 32W05, 35A27. Secondary 32C05, 32T25, 46E25.

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Received 19 April 2021

Accepted 12 January 2022

Published 13 May 2022