Sobolev Estimates for Fractional and Singular Radon Transforms

We prove Sobolev inequalities for singular and fractional Radon transforms which are defined as in a paper of Phong and Stein under certain hypothesis on the corresponding Lagrangian ((N*C)′) which does not necessarily have to be a canonical graph. In the proof we use oscillatory integrals, the Cotlar–Stein almost orthogonality theorem, a sort of Littlewood–Paley decomposition for a certain operator, some basic facts about Fourier integral operators and pseudodifferential operators. The main ideas come from papers by Phong and Stein (Acta Math.157(1986), 99–157) and Sogge and Stein (J. Analyse Math.54(1990), 165–188).