Fractional integrals associated with Radon transforms

Abstract

Abstract We obtain sharp L p {L^{p}} - L q {L^{q}} estimates for fractional integrals generated by Radon transforms of the following three types: The classical Radon transform over the set of all hyperplanes in ℝ n {\mathbb{R}^{n}} , the Strichartz transversal transform over only those hyperplanes which meet the last coordinate axis, and the Radon transform associated with paraboloids. The method relies on a version of Stein’s interpolation theorem for analytic families of operators communicated by L. Grafakos.