Lp-Lq-estimates for potential-type operators with oscillating kernels

We consider a class of multidimensional potential-type operators whose kernels are oscillating at infinity. The characteristics of these operators are from a wide class of functions including the product of a homogeneous function infinitely differentiable in Rn∖{0} and any function from Cm,γ(R˙1+). We describe convex sets in the (1/p;1/q)-plane for which these operators are bounded from Lp into Lq and indicate the domains where they are not bounded. In some cases, the accuracy of the estimates obtained is proved. In particular, necessary and sufficient conditions for the boundedness of the operators under considered in Lp are obtained. Currently, there is a number of papers on Lp-Lq-estimates for convolution operators with oscillating kernels, in particular, for the Bochner-Riesz operators and acoustic potentials arising in various problems of analysis and mathematical physics. These papers cover kernels containing only the radial characteristic b(r), which stabilized at infinity as a Helder function...