On a Chain of Harmonic and Monogenic Potentials in Euclidean Half-space

TL;DR

This paper constructs a sequence of harmonic and monogenic potentials in Euclidean half-space, generalizing complex functions, and explores their boundary limits, connecting them to classical distributions and operators in Clifford analysis.

Contribution

It introduces a chain of potentials in higher dimensions, extending complex logarithmic functions, and links boundary limits to well-known distributions and operators.

Findings

Boundary limits are classical distributions like Dirac and Hilbert kernels.

Potentials can be recovered via convolution with boundary distributions.

The work generalizes complex analysis concepts to higher-dimensional Clifford analysis.

Abstract

In the framework of Clifford analysis, a chain of harmonic and monogenic potentials is constructed in the upper half of Euclidean space $\mR^{m+1}$, including a higher dimensional generalization of the complex logarithmic function. Their distributional limits at the boundary $\mR^{m}$ turn out to be well-known distributions such as the Dirac distribution, the Hilbert kernel, the fundamental solution of the Laplace and Dirac operators, the square root of the negative Laplace operator, and the like. It is shown how each of those potentials may be recovered from an adjacent kernel in the chain by an appropriate convolution with such a distributional limit.