On the convolution product of the distributional kernel Kα,β,γ,ν

Abstract

We introduce a distributional kernel Kα,β,γ,ν which is related to the operator ⊕ k iterated k times and defined by ⊕ k=[(∑r=1p∂2/∂xr2)4−(∑j=p+1p+q∂2/∂xj2)4] k, where p+q=n is the dimension of the space ℝ n of the n-dimensional Euclidean space, x=(x1,x2,…,xn)∈ℝ n, k is a nonnegative integer, and α, β, γ, and ν are complex parameters. It is found that the existence of the convolution Kα,β,γ,ν∗Kα′,β′,γ′,ν′ is depending on the conditions of p and q.