Weighted norm estimates for the Semyanistyi fractional integrals and Radon transforms

Abstract

Semyanistyi's fractional integrals have come to analysis from integral geometry. They take functions on R n to functions on hyperplanes, commute with rotations, and have a nice behavior with respect to dilations. We obtain sharp inequalities for these integrals and the corresponding Radon transforms acting on L p spaces with a radial power weight. The operator norms are explicitly evaluated. Similar results are obtained for fractional integrals associated to k-plane transforms for any 1 ≤ k < n.