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

Abstract

Explicit inversion formulas of Balakrishnan–Rubin type and a characterization of Bessel potentials associated with the Laplace–Bessel differential operator \(\Delta _B = \sum\limits_{k = 1}^{n - 1} {\frac{{\partial ^2 }}{{\partial x_k^2 }}} + \left( {\frac{{\partial ^2 }}{{\partial x_n^2 }} + \frac{{2\nu }}{{x_n }}\frac{{\partial ^2 }}{{\partial x_n^2 }}} \right){\text{ }}\left( {\nu > 0} \right)\) are obtained. As an auxiliary tool the B-metaharmonic semigroup is introduced and some of its properties are investigated.