Abstract

In this paper, we study the elementary solution of the operator
$\circledast_B^k$ which is defined by
$$\circledast_B^k=\left[\left(B_{x_1}+B_{x_2}+\cdots+B_{x_p}\right)^3
+\left(B_{x_{p+1}}+\cdots+B_{x_{p+q}}\right)^3\right]^k,$$ where
$p+q=n$ is the dimension of
$\mathbb{R}^+_n=\{(x=x_1,x_2,\dots,x_n):x_1>0,x_2>0,\dots,x_n>0\}$,
$B_{x_i}=\frac{\partial^2}{\partial x_i^2}+
\frac{2v_i}{x_i}\frac{\partial}{\partial x_i}$,
$2v_i=2\alpha_i+1$, $\alpha_i>-\frac{1}{2}$, $x_i>0$,
$i=1,2,\dots,n$ and $k$ is a positive integer. After that, we
apply such an elementary solution to solve the equation
$\circledast_B^ku(x)=f(x)$, where $f$ is a generalized function
and $u$ is an unknown function.

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