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

2012/10/07 by Fred Brackx, Brackx, Fred, Hendrik De Bie +3 · 2 citations
Mathematics · #30G35 #31C45 #Advanced Algebra and Geometry #Algebraic and Geometric Analysis #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Analysis and Transform Methods #math.CA #msc:30G35 #msc:31C45

paper · pdf · doi:10.48550/arxiv.1210.2044

arxiv created 2012/10/07 · openalex publication_date 2012/10/07 · arxiv updated 2012/10/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Cited by

• Distributional Boundary Values of Harmonic Potentials in Euclidean Half-space as Fundamental Solutions of Convolution Operators in Clifford Analysis
• Harmonic and Monogenic Potentials in Low Dimensional Euclidean Half-Space

Related

• Representation of Distributions by Harmonic and Monogenic Potentials in Euclidean Space
• Distributional Boundary Values of Harmonic Potentials in Euclidean Half-space as Fundamental Solutions of Convolution Operators in Clifford Analysis
• Harmonic and Monogenic Potentials in Low Dimensional Euclidean Half-Space
• Two Families of Clifford Distributions Revisited
• Fourier Spectrum Characterizations of Clifford H^p Spaces on Rⁿ⁺¹₊ for 1≤ p ≤ ∞