The distributional products on spheres and Pizetti’s formula

The distribution δ(k)(r−a)δ(k)(r−a) concentrated on the sphere OaOa with r−a=0r−a=0 is defined as (δ(k)(r−a),ϕ)=(−1)kan−1∫Oa∂k∂rk(ϕrn−1)dσ. Taking the Fourier transform of the distribution and the integral representation of the Bessel function, we obtain asymptotic expansions of δ(k)(r−a)δ(k)(r−a) for k=0,1,2,…k=0,1,2,… in terms of △jδ(x1,…,xn)△jδ(x1,…,xn), in order to show the well-known Pizetti formula by a new method. Furthermore, we derive an asymptotic product of ϕ(x1,…,xn)δ(k)(r−a), where ϕϕ is an infinitely differentiable function, based on the formula of △m(ϕψ)△m(ϕψ), and hence we are able to characterize the distributions focused on spheres, which can be written as the sums of multiplet layers in the Gel’fand sense.