On higher order Bessel potentials

Abstract

The solutions of the equation \( -(\Delta - I)^{k}u_{k} = f, k \geq 1 \) in \( \mathbb{R}^n \) , where \( f \in L^{2} (\mathbb{R}^n) \) are investigated, Bessel potentials of higher order are defined, and recurrence relations between these solutions and these Bessel potentials are obtained. It is also proved that these solutions and the solutions of \( -\Delta^{k}v_{k} + v_{k} = \varrho \), under certain conditions, are identical.