Effective Kohn Algorithm for Special Domain Defined by Functions Depending on All Variables

2023/06/28 by Yum-Tong Siu, Siu, Yum-Tong
Computer Science · Mathematics · #13H15 (Secondary #32T25 (Primary) 32T27 #35N15 #Advanced Mathematical Modeling in Engineering #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2306.16494

openalex publication_date 2023/06/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Kohn introduced in 1979 the algorithm of multipliers to study the subelliptc estimate of the ∂-Neumann problem for a smooth weakly pseudoconvex domain in a complex Euclidean space which satisfies D'Angelo's finite type condition of a finite bound for the normalized touching order to the boundary for any local possibly singular holomorphic curve in the complex Euclidean space. The problem can be regarded as an example of the formulation of Hörmander's 1967 hypoelliptic result for the case of complex-valued vector-valued unknowns. So far the effective solution of Kohn's problem is known only for special domains \rm Re(z_n+1)+∑_j=1^N|f_j|²<0 in \mathbb Cⁿ⁺¹ with f_j holomorphic in z₁,...,z_n, because for such domains it suffices to deal with holomorphic multipliers. One main obstacle to treat the general smooth case is the need to deal with nonholomorphic multipliers. This note introduces a new technique to handle nonholomorphic multipliers occurring in more general domains with f_j holomorphic in all the n+1 variables z₁,...,z_n,z_n+1.

Related

• Explicit calculation of Siu's Effective Termination in Kohn's Algorithm for Special Domains in ℂ³
• Effective Termination of Kohn's Algorithm for Subelliptic Multipliers
• Complex Neumann Problem and Kohn's Algorithm
• The Kohn-Hörmander-Morrey formula twisted by a pseudodifferential operator
• Hypoellipticity of the ∂-Neumann problem by means of subelliptic multipliers