The solutions of the $n$-dimensional Bessel diamond operator and the Fourier--Bessel transform of their convolution

Abstract

In this article, the operator $\Diamond_{B}^{k}$ is introduced and named as the Bessel diamond operator iterated $k$ times and is defined by $ \Diamond_{B}^{k} = [ (B_{x_{1}} + B_{x_{2}} + ... + B_{x_{p}})^{2} - (B_{x_{p + 1}} + ... + B_{x_{p + q}})^{2} ]^{k}$, where $ p + q = n, B_{x_{i}} = \frac{\partial^{2}}{\partial x_{i}^{2}} + \frac{2v_{i}}{x_{i}} \frac{\partial}{\partial x_{i}}, $ where $2v_{i} = 2\alpha_{i} + 1$, $ \alpha_{i} > - {1/2} $ [8], $x_{i} > 0$, $i = 1, 2, ..., n, k$ is a non-negative integer and $n$ is the dimension of $\mathbb{R}_{n}^{+}$. In this work we study the elementary solution of the Bessel diamond operator and the elementary solution of the operator $\Diamond_{B}^{k}$ is called the Bessel diamond kernel of Riesz. Then, we study the Fourier--Bessel transform of the elementary solution and also the Fourier--Bessel transform of their convolution.