The products on the unit sphere and even-dimension spaces

The distribution δ(k)(r−1) focused on the unit sphere Ω of Rm is defined by 〈δ(k)(r−1),ϕ〉=(−1)k∫Ω∂k∂rk(ϕrm−1)dω, where ϕ is Schwartz testing function. We apply the expansion formula ∫Ω∂k∂rkϕ(rω)dω=(−1)k〈∑i=0k(ki)C(m,i)δ(k−i)(r−1),ϕ(x)〉 to evaluate the product of f(r) and δ(k)(r−1) on Ω. Furthermore, utilizing the Laurent series of rλ and the residue of 〈rλ,ϕ〉 at the singular point λ=−m−2k, we derive that δ2(x)=0 on even-dimension space. Finally, we are able to imply Δk(r2k−mlnr)⋅δ(x)=0 based on the fact that r2k−mlnr is an elementary solution of partial differential equation ΔkE=δ(x) by using the generalized Fourier transform.