On the solutions of the n-dimensional diamond operator

摘要

In this paper, we consider the solution of the equation where is introduced and named as the diamond operator interated k times and is defined by Now x = (x1, x2,…, xn) ϵ Rn, the n-dimensional Euclidian space, p + q = n, cr is a constant, δ is the Dirac-delta distribution, and and k = 0, 1, 2, 3,….It is found that the type of solutions of this equation, such as the ordinary functions, the tempered distributions and the singular distributions, depend on the relationship between the values of k and m.