Regularization and derivatives of multipole potentials

Abstract

The only harmonic homogeneous functions defined in Rn∖{0} are the harmonic polynomials and the so-called multipole potentials, namely functions of the type P(x)=p(x)/|x|2k+n−2 for some harmonic polynomial p of degree k. The first aim of this article is to study the distributional regularization of multipole potentials. We show that even though the Hadamard regularization Pf(p(x)/|x|2k+n−2) exists for any homogeneous polynomial of degree k, the principal value p.v.(p(x)/|x|2k+n−2) exists if and only if p is harmonic; this means that if p is harmonic then for any test function ϕ the divergent integral ∫Rnp(x)ϕ(x)/|x|2k+n−2dx can be computed by employing polar coordinates and performing the angular integral first. We also find the first and second order distributional derivatives of these regularizations and, more generally, of the regularizations of functions of the form Pl(x)=p(x)/|x|k+l. We find many interesting formulas that hold precisely when p is a harmonic polynomial of degree k. In particular, we prove thatΔ‾p.v.(p(x)r2k+n−2)=(−1)k+1πn/22k−2Γ(n2+k−1)p(∇)δ(x), generalizing the well known relation Δ‾(r2−n)=(2−n)Cδ(x), where C is the area of a sphere of radius 1. Actually formulas like this one hold for a homogeneous polynomial p of degree k if and only if p is harmonic.