Lp-Lp′ estimates for overdetermined radon transforms

We prove several variations on the results of F. Ricci and G. Travaglini (2001), concerning L-p - L-p' bounds for convolution with all rotations of arc length measure on 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 Ricci and Travaglini's approach, 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 the two techniques.