Operators associated to the Cauchy-Riemann operator in elliptic complex numbers

TL;DR

This paper generalizes the solvability of first order systems involving the Cauchy-Riemann operator to elliptic complex numbers with arbitrary parameters, extending classical results to broader contexts.

Contribution

It extends the Son-Tutschke lemma to a wider class of parameters, including non-elliptic cases, and develops interior estimates for elliptic cases using generalized Cauchy representation.

Findings

Validates the Son-Tutschke lemma for a large class of parameters

Provides first order interior estimates in elliptic complex numbers

Solves initial value problems with holomorphic functions in elliptic complex numbers

Abstract

In this article we provide a generalized version of the result of L.H. Son and W. Tutschke \cite{tut} on the solvability of first order systems on the plane whose initial functions are arbitrary holomorphic functions. This is achieved by considering the more general concept of holomorphicity with respect to the structure polynomial $X^2 + \beta X + \alpha$. It is shown that the Son-Tutschke lemma on the construction of complex linear operators associated to the Cauchy-Riemann operator remains valid when interpreted for a large class of real parameters $\alpha$ and $\beta$ including the elliptic case but also cases that are not elliptic. For the elliptic case, first order interior estimates are obtained via the generalized version of the Cauchy representation theorem for elliptic numbers and thus the method of associated operators is applied to solve initial value problems with initial functions that are holomorphic in elliptic complex numbers.