Distributional Boundary Values of Harmonic Potentials in Euclidean Half-space as Fundamental Solutions of Convolution Operators in Clifford Analysis

TL;DR

This paper explores the boundary behavior of harmonic potentials in Clifford analysis, revealing their connection to fundamental pseudodifferential operators and extending classical potential theory to higher dimensions.

Contribution

It introduces a new understanding of boundary limits of harmonic and monogenic potentials in Clifford analysis and relates them to key pseudodifferential operators.

Findings

Distributional boundary values linked to Dirac and Laplace operators

Extension of harmonic potential theory to higher dimensions

Identification of boundary limits as fundamental solutions

Abstract

In the framework of Clifford analysis, a chain of harmonic and monogenic potentials in the upper half of Euclidean space $\mR^{m+1}$ was recently constructed, including a higher dimensional analogue of the logarithmic function in the complex plane. In this construction the distributional limits of these potentials at the boundary $\mR^{m}$ are crucial. The remarkable relationship between these distributional boundary values and four basic pseudodifferential operators linked with the Dirac and Laplace operators is studied.