A Quaternionic Beltrami-Type Equation and the Existence of Local Homeomorphic Solutions

Abstract

The paper deals with a quaternionic Beltrami-type equation, which is a very natural generalization of the complex Beltrami equation to higher dimensions. Special attention is paid to the systematic use of the embedding of the set of quaternions $\mathbb{H}into{\mathbb{C}}^2$ and the corresponding application of matrix singular integral operators. The proof of the existence of local homeomorphic solutions is based on a necessary and sufficient criterion, which relates the Jacobian determinant of a mapping from ${\mathbb{R}}^4$ into ${\mathbb{R}}^4$ to the quaternionic derivative of a monogenic function.