On the \(B\)-convolutions of the Bessel diamond kernel of Riesz
Let \(\Delta_B=\sum_{i=1}^nB_{x_i} =\sum_{i=1}^n\Big(\frac{\partial^2}{\partial x_i^2}+\frac{2\nu_i}{x_i}\frac{\partial}{\partial x_i}\Big)\) be the Laplace-Bessel operator and \(\square_B=B_{x_1}+\cdots+B_{x_p}-B_{x_{p+1}}-\cdots-B_{x_{p+q}}\), \(p+q=n\) the ultra-hyperbolic type operator. Then the Bessel diamond operator iterated \(k\)-times \(\diamond_B^k\) is given by \(\square_B^k\Delta_B^k\) and the elementary solution is given by \(T_k=(-1)^kS_{2k}*R_{2k}\), which is called the Bessel diamond kernel of Riesz, where \((-1)^kS_{2k}\) and \(R_{2k}\) are unique elementary solutions \(\Delta_B^k\) and \(\square_B^k\) respectively and \(*\) is the \(B\)-convolution associated to \(\diamond_B^k\). The authors investigate the \(B\)-convolution structure of \(T_k\) as in the classical case of \(\nu_i=0\).
scientific article; zbMATH DE number 5522153