Complex associated to some systems of PDE

TL;DR

This paper constructs the Cauchy-Fueter complex in ^8 using Cartan's theory, providing conditions for when such complexes contain only first-order operators, with the Cauchy-Fueter case as an example.

Contribution

It offers a direct construction of the Cauchy-Fueter complex in ^8 and establishes conditions for complexes of linear PDEs with constant coefficients to have only first-order operators.

Findings

Constructed the Cauchy-Fueter complex in ^8.

Provided a sufficient condition for complexes to contain only first-order operators.

Showed the Cauchy-Fueter equation in ^8 does not satisfy this condition.

Abstract

In [WW1] and [WW2], the author constructed the complex associated to 1-regular functions. This complex is the equivalent of Dolbeault's complex for holomorphic functions if we replace the Cauchy-Riemann equations by the Cauchy-Fueter equations. In this paper, using the Cartan theory of linear Pfaffian system, we give a direct construction for the Cauchy-Fueter complex, at least in $\R^8$. Moreover, we give a sufficient condition in terms of Cartan's theory, to ensure that a complex associated to a linear PDE system with constant coefficients of order one, contains only operators of order one. In fact, the Cauchy-Fueter equation in $\R^8$ is an illuminating example for which this condition is not satisfied