An n-dimensional pseudo-differential operator involving the Hankel transformation

An n-dimensional pseudo-differential operator (p.d.o.) involving the n-dimensional Hankel transformation is defined. The symbol class Hm is introduced. It is shown that p.d.o.’s associated with symbols belonging to this class are continuous linear mappings of the n-dimensional Zemanian space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H_\mu(I^n)$\end{document} into itself. An integral representation for the p.d.o. is obtained. Using the Hankel convolution, it is shown that the p.d.o. satisfies a certain L1-norm inequality.