Summary: We obtain \(L_p\to L_p\)-estimates for the potential operator \(K_a^\alpha\) in \(\mathbb{R}^n\) with a radial kernel of the form \(a(| t|) e^{i| t|}/ | t|^{n-\alpha}\), \(0<\alpha <n\), where \(a(\infty)\neq 0\) and the characteristic \(a(r)\) is sufficiently smooth in some neighborhood of infinity and locally satisfies some general assumptions. In particular, \(a(r)\) may have power singularities. We construct some convex sets on the \((1/p,1/q)\)-plane for which the operator \(K_a^\alpha\) is bounded from \(L_p\) into \(L_q\), as well as the domains where \(K_a^\alpha\) is not bounded.