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 ∂rk φ(rω)dω = (−1)k k i=0 k i 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−m ln r) · δ(x) = 0 based on the fact that r2k−m ln r is an elementary solution of partial differential equation ΔkE = δ(x) by using the generalized Fourier transform.  2004 Elsevier Inc. All rights reserved