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

TL;DR

This paper advances the Kohn algorithm for special domains by developing a new technique to handle nonholomorphic multipliers, extending its applicability beyond previously solvable cases.

Contribution

It introduces a novel method to manage nonholomorphic multipliers in the Kohn algorithm for more general domains, broadening the scope of effective solutions.

Findings

Successfully handles nonholomorphic multipliers in complex domains

Extends the effective Kohn algorithm to more general settings

Provides a new technique for complex analysis in PDEs

Abstract

Kohn introduced in 1979 the algorithm of multipliers to study the subelliptc estimate of the $\bar\partial$-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 H\"ormander'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 ${\rm Re}(z_{n+1})+\sum_{j=1}^N|f_j|^2<0$ in ${\mathbb C}^{n+1}$ with $f_j$ holomorphic in $z_1,...,z_n$, 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 $z_1,...,z_n,z_{n+1}$.