Lp-Lp' estimates for overdetermined radon transforms

We prove several variations on the results in Ricci and Travaglini [Convex curves, Radon transforms and convolution operators defined by singular measures, [Proc. Amer. Math. Soc. v.129 (2001) pp. 1739-1744] concerning L^p-L^{p'} bounds for convolution with all rotations of a measure supported by a fixed convex curve in R^2. Estimates are obtained for averages over higher-dimensional convex (nonsmooth) hypersurfaces, smooth k-dimensional surfaces, and nontranslation-invariant families of surfaces. We compare the approach of Ricci and Travaglini, based on average decay of the Fourier transform, with an approach based on L^2 boundedness of Fourier integral operators, and show that essentially the same geometric condition arises in proofs using different techniques.