On the convolutions of the diamond kernel of Marcel Riesz

摘要

In this paper, we consider the equation ♢ku(x)=δ where ♢k is introduced and named as the diamond operator iterated k-times and is defined by ♢k=((∑pi=1(∂2/∂x2i))2−(∑p+qj=p+1(∂2/∂x2j))2)k,u(x) is a generalized function, x=(x1,x2,…,xn)∈Rn the n-dimensional Euclidean space, p+q=n, k=0,1,2,3,…andδ is the Dirac-delta distribution. Now u(x) is the elementary solution of the operator ♢k and is called the diamond kernel of Marcel Riesz. The main part of this work is studying the convolution of u(x).