A Cauchy--Riemann-Type CK Extension for Monogenic Functions

Abstract

We introduce a generalized Cauchy--Kowalevski extension associated with a biaxial decomposition of ${\mathbb{R}}^{m+1}$, allowing monogenic functions to be reconstructed from data prescribed on the subspace $\{X\in {\mathbb{R}}^{m+1}:z=0\}$ where $z=x_0+x_1{e}_1$. The extension admits a natural decomposition into monogenic layers, leading to operators ${\mathrm{CK}}_{z,k}$. Classes of biaxial monogenic functions generated by these operators are characterized by systems of partial differential equations, and explicit power series solutions are obtained. The action of the operators ${\mathrm{CK}}_{z,k}$ on homogeneous polynomials is then investigated, leading to a class of biaxial monogenic plane waves and the associated system of partial differential equations.

16 pages