On the B-convolutions of the Bessel diamond kernel of Riesz

In this article, the operator ♢Bk is introduced and named as the Bessel diamond operator iterated k-times and is defined by♢Bk=[(Bx1+Bx2+⋯+Bxp)2-(Bxp+1+⋯+Bxp+q)2]kwhere p+q=n,Bxi=∂2∂xi2+2vixi∂∂xi, 2vi=2αi+1, αi>-12 [8], xi>0,i=1,2,…,n,k is a nonnegative integer and n is the dimension of the Rn+. In this work, we study the elementary solution of the operator ♢Bk and this elementary solution is called the Bessel diamond kernel of Riesz. Then, we study the B-convolution of this elementary solution.