On the convolution equation related to the diamond kernel of Marcel Riesz (Q1298667)

Let \[ \diamondsuit^k = \Biggl[ \biggl({\partial^2\over \partial t^2_1} + {\partial^2\over \partial t^2_2}+\cdots +{\partial^2\over \partial t^2_p} \biggr)^2 - \biggl({\partial^2\over \partial t^2_{p+1}} + {\partial^2\over \partial t^2_{p+2}}+\cdots +{\partial^2\over \partial t^2_{p+q}}\biggr)^2 \Biggr]^k,\;p+q=n,\;t\in {\mathbb R}^n \] be the diamond operator iterated \(k\)-times (\(k=0,1,2,\dots\)). In the paper the author first studies properties of the distribution \(e^{\alpha t} \diamondsuit^k \delta\), \(\alpha,t\in {\mathbb R}\). As an application he considers the convolution equation \[ (e^{\alpha t} \diamondsuit^k \delta)*u(t) = e^{\alpha t}\sum_{r=0}^m c_r \diamondsuit^k \delta, \] giving explicit formulas for the elementary solution of this equation depending on \(m, k\) and \(\alpha\).