On the Convolution Equation Related to the Diamond Klein-Gordon Operator

Abstract

We study the distribution eαx(♢+m2)kδ for m≥0, where (♢+m2)k is the diamond Klein-Gordon operator iterated k times, δ is the Dirac delta distribution, x=(x1,x2,…,xn) is a variable in ℝn, and α=(α1,α2,…,αn) is a constant. In particular, we study the application of eαx(♢+m2)kδ for solving the solution of some convolution equation. We find that the types of solution of such convolution equation, such as the ordinary function and the singular distribution, depend on the relationship between k and M.