On the Fourier transform of the diamond kernel of marcel Riesz (Q1294246)

In his earlier papers the author found an explicit formula for the elementary solution of the equation \[ \diamondsuit^k u(x) =\delta, \] where \[ \begin{multlined} \diamondsuit^k = \Biggl[\biggl( {\partial^2\over \partial x^2_1} + {\partial^2\over \partial x^2_2}+\cdots +{\partial^2\over \partial x^2_p} \biggr)^2 - \biggl( {\partial^2\over \partial x^2_{p+1}} + {\partial^2\over \partial x^2_{p+2}}+\cdots +{\partial^2\over \partial x^2_{p+q}}\biggr)^2 \Biggr]^k,\\ p+q=n,\;x\in {\mathbb R}^n \end{multlined} \] is the diamond operator iterated \(k\)-times (\(k=0,1,2,\dots\)). This solution (called the diamond kernel of Marcel Riesz) was given in the form of convolution \(u(x) = (-1)^k S_{2k}(x)*R_{2k}(x)\), where both \(S_\alpha\) and \(R_\alpha\) are ordinary functions for \(\alpha \geq n\) and distributions if \(\alpha < n\). In the paper the author continues to study properties of the diamond kernel \((-1)^k S_{2k}(x)*R_{2k}(x)\) giving the exact formulae for its Fourier transform and also for the Fourier transform of its convolutions.