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). (C) 1996 Academic Press, Inc.