The Laplacian and Related Operators
Starting from first principles, all fundamental solutions (that are tempered distributions) for scalar elliptic operators are identified in this chapter. While the natural starting point is the Laplacian, this study encompasses a variety of related operators, such as the bi-Laplacian, the poly-harmonic operator, the Helmholtz operator and its iterations, the Cauchy–Riemann operator, the Dirac operator, the perturbed Dirac operator and its iterations, as well as general second-order constant coefficient strongly elliptic operators. Having accomplished this task then makes it possible to prove the well-posedness of the Poisson problem (equipped with a boundary condition at infinity), and derive qualitative/quantitative properties for the solution. Along the way, Cauchy-like integral operators are also introduced and their connections with Hardy spaces are brought to light in the setting of both complex and Clifford analyses.