$L^{p}-L^{p'}$ estimates for overdetermined Radon transforms

We prove several variations on the results of F. Ricci and G. Travaglini (2001), concerning $L^{p}-L^{p'}$ bounds for convolution with all rotations of arc length measure on a fixed convex curve in $\mathbb{R} ^{2}$. 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.