On Inversion Of Bessel Potentials Associated With The Laplace–Bessel Differential Operator

Explicit inversion formulas of Balakrishnan–Rubin type and a characterization of Bessel potentials associated with the Laplace–Bessel differential operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\Delta _B = \sum\limits_{k = 1}^{n - 1} {\frac{{\partial ^2 }}{{\partial x_k^2 }}} + \left( {\frac{{\partial ^2 }}{{\partial x_n^2 }} + \frac{{2\nu }}{{x_n }}\frac{{\partial ^2 }}{{\partial x_n^2 }}} \right){\text{ }}\left( {\nu > 0} \right)$$ \end{document} are obtained. As an auxiliary tool the B-metaharmonic semigroup is introduced and some of its properties are investigated.