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

Abstract

In this article, the operator$\diamondsuit _B^k$ is introduced and named as the Bessel diamond operator iteratedk times and is defined by

$$\diamondsuit _B^k = [(B_{x_1 } + B_{x_2 } + \cdots + B_{x_p } )^2 - (B_{x_{p + 1} } + \cdots + B_{x_{p + q} } )^2 ]^k $$

$p + q = n,B_{x_i } = \tfrac{{\partial ^2 }}{{\partial x_i^2 }} + \tfrac{{2v_i }}{{x_i }}\tfrac{\partial }{{\partial x_i }}$ where$2v_i = 2\alpha _i + 1,\alpha _i > - \tfrac{1}{2}[8],x_i > 0$,i = 1, 2, ...,n k is a non-negative integer andn is the dimension of ℝ ⁺_n . In this work we study the elementary solution of the Bessel diamond operator and the elementary solution of the operator$\diamondsuit _B^k$ 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.