Harmonic and Monogenic Potentials in Low Dimensional Euclidean Half-Space

TL;DR

This paper computes harmonic and monogenic potentials and their boundary values in 3 and 4 dimensions within Clifford analysis, extending previous higher-dimensional results and addressing dimensional-specific challenges.

Contribution

It provides explicit calculations of potentials and boundary values in dimensions 3 and 4, where general formulas do not apply, advancing the understanding of Clifford analysis in low dimensions.

Findings

Explicit potentials in 3 and 4 dimensions derived

Boundary values at the boundary space computed

Addresses dimensional-specific computational challenges

Abstract

In the framework of Clifford analysis, a chain of harmonic and monogenic potentials in the upper half of Euclidean space R^{m+1} was constructed recently, including a higher dimensional analogue of the logarithmic function in the complex plane. Their distributional limits at the boundary R^{m} were also determined. In this paper the potentials and their distributional boundary values are calculated in dimensions 3 and 4, dimensions for which the expressions in general dimension break down.