L(p) - L(p') estimates for overdetermined Radon transforms.

We prove several variations on the results of F. Ricci and G. Travaglini (2001), concerning L p − L p ′ L^{p}-L^{p'} bounds for convolution with all rotations of arc length measure on a fixed convex curve in R 2 \mathbb {R} ^{2} . Estimates are obtained for averages over higher-dimensional convex (nonsmooth) hypersurfaces, smooth k 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 L^{2} boundedness of Fourier integral operators, and show that essentially the same geometric condition arises in proofs using the two techniques