Representation of distributions by harmonic and monogenic potentials in Euclidean space

In the framework of Clifford analysis, a chain of harmonic and monogenic potentials in the upper half of Euclidean space \({\mathbb{R}^{m+1}_+}\) was recently constructed, including a higher dimensional analogue of the logarithmic function in the complex plane, and their distributional boundary values were computed. In this paper we determine these potentials in lower half–space \({\mathbb{R}^{m+1}_-}\) and investigate whether they can be extended through the boundary \({\mathbb{R}^m}\) . This is a stepping stone to the representation of a doubly infinite sequence of distributions in \({\mathbb{R}^m}\) , consisting of positive and negative integer powers of the Dirac and the Hilbert–Dirac operator, as the jump across \({\mathbb{R}^m}\) of monogenic functions in the upper and lower half–spaces, in this way providing a sequence of interesting examples of Clifford hyperfunctions.