On higher order Bessel potentials (Q1881510)

The authors investigate the solutions of the equation \(-(\Delta-I)^k u_k=f,\) \(k\geq1,\) in \(\mathbb R^n,\) where \(f\in L^2(\mathbb R^n).\) These lead to the definition of higher-order Bessel potentials through precise recurrence relations between \(u_k\)'s. Moreover, it is proved that \(u_k\)'s coincide with the solutions \(v_k\) of \(-\Delta^k v_k+v_k=\rho,\) where \[ f=\rho +\sum_{\nu=1}^{k-1} \left( k \atop \nu\right) u_\nu, \] \(-(\Delta-I) u_1=f\) and \((\Delta-I) u_\nu=u_{\nu-1}\) for \(2\leq\nu\leq k-1.\)