On generalized Riesz type potential with Lorentz distance

In this article, we defined the Bessel ultra-hyperbolic operator iterated k—times and is defined by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \square _B^k = \left[ {B_{x1} + B_{x2} + \cdots + B_{x_p } - B_{x_{p + 1} } - \ldots - B_{x_{p + q} } } \right]^k $$\end{document}, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ p + q = n, B_{x_i } = \frac{{\partial ^2 }} {{\partial x_i^2 }} + \frac{{2v_i }} {{x_i }}\frac{\partial } {{\partial x_i }}, 2v_i = 2\alpha _i + 1,\alpha _i > - \frac{1} {2} [4] $$\end{document}, xi>0, i=1,2, ..., n, k is a nonnegative integer and n is the dimension of the ℝn1. Furthermore we have generated the generalized ultra-hyperbolic Riesz potential with Lorentz distance. This potential is generated by the generalized shift operator for functions in Schwartz spaces.