The Solution of the -Dimensional of the Operator D_{B,C}^k

In this paper,  the operator $D_{B,C}^k$ is the partial differential operators related to the diamond Bessel operator  iterated $k$-times and is defined by $$D_{B,C}^{k}= \left(\frac{1}{C^4}\left(\sum_{i=1}^{p} B_{x_i}\right)^2- \left( \sum_{j=p+1}^{p+q} B_{x_j}\right)^2 \right)^k,$$ 

where $p+q=n, x=(x_{1},\ldots , x_{n})\in \mathbb{R}^{+}_n, B_{x_{i}}=\frac{\partial ^{2}}{\partial x_{i}^{2}}+ \frac{2v_{i}In this paper, the operator is the partial differential operators related to the diamond Bessel operator iterated -times and is defined by where , , is a nonnegative integer and is the dimension of the , is a constant, is a positive real number, is the Dirac-delta distribution, . It is shown that, depending on the relationship between and , the solution to this equation can be ordinary functions, tempered distributions, or singular distributions. }{x_{i}}\frac{\partial }{\partial x_{i}}, v_{i}=2\alpha _{i}+1, \alpha _{i}>-\frac{1}{2}\;\;\cite{Levitan},  x_{i}>0, i=1,2,\ldots,n, a_{r}$ is a constant, $C$ is a positive real number, $k$ is a nonnegative integer, $\delta$  is the Dirac-delta distribution, $D_{B,C} ^{0}\delta =\delta$  and $n$ is the dimension of $\mathbb{R}^{+}_n$. It is shown that, depending on the relationship between $k$ and $m$, the solution to this equation can be ordinary functions, tempered distributions, or singular distributions.