Hyperbolic hypercomplex D-bar operators, hyperbolic CR-equations and harmonicity

TL;DR

This paper reviews developments in hyperbolic hypercomplex analysis, introduces new hyperbolic harmonicity concepts, and systematically develops $ar{ ext{d}}$-operator theory for generalized hypercomplex algebras, including microlocal analysis of solutions.

Contribution

It provides a systematic $ar{ ext{d}}$-operator framework for hypercomplex algebras and explores hyperbolic harmonicity with microlocal analysis, extending prior hypercomplex analysis work.

Findings

Development of $ar{ ext{d}}$-operator theory for hypercomplex algebras

Introduction of hyperbolic harmonicity concepts

Microlocal analysis of PDE solutions in hypercomplex settings

Abstract

This paper is partially a review of the development of the Investigation Program announced by Stancho Dimiev at the Bedlevo Conference on Hypercomplex Analysis (2006). A new aspect related with hyperbolic complex numbers, their generalizations and applications on hyperbolic harmonicity is included in some Appendix of S. Dimiev. The recent results on the hyperbolic complex structures are exposed. We describe systematically the $\partial$-bar trik which led to Cauchy-Riemann type systems for different generalized hypercomplex algebras. This is in fact an elementary $\partial$-bar type theory of the PDE on the mentioned algebras. Microlocal treatment (distributions and Fourier transform) of the solutions is included too, with one result of P. Popivanov.