(untitled) (Analysis in Theory and Applications, 2018)
It is well-known that if p is a homogeneous polynomial of degree k in n variables, p∈P_k, then the ordinary derivative p(∇) (r^2−n) has the form A_n,kΥ(x)r^2−n−2k where A_n,k is a constant and where Υ is a harmonic homogeneous polynomial of degree k, ΥH_k, actually the projection of p onto H_k. Here we study the distributional derivative p(∇)(r^2−n) and show that the ordinary part is still a multiple of Υ, but that the delta part is independent of Υ, that is, it depends only on p−Υ. We also show that the exponent 2−n is special in the sense that the corresponding results for p(∇)(r^α) do not hold if α eq 2−n. Furthermore, we establish that harmonic polynomials appear as multiples of r^2−n−2k−2k' when p(∇) is applied to harmonic multipoles of the form Υ'(x)r^2−n−2k' for some Υ' H_k.