L_p-l_q-estimates for generalized Riss potentials with oscillating kernels

We consider a class of multidimensional potential-type operators whose kernels are oscillating at infinity. The characteristics of these operatorsare infinitely differentiable homogeneous functions. We describe convex setsin the $(1/p;1/q)$-plane for which these operators are bounded from $L_p$into $L_q$ and indicate the domains where they are not bounded. In some caseswe describe their $\cal{L}$-characteristics. To obtain these results we usea new method based on special representation of the symbols of multidimensionalpotential-type operators. To these representations of the symbols we apply the techniqueof Fourier-multipliers, which degenerate or have singularities on the unitsphere in $\mathbb{R}^n$.