𝐿^{𝑝}-𝐿^{𝑝’} estimates for overdetermined Radon transforms

Author:

Brandolini Luca,Greenleaf Allan,Travaglini Giancarlo

Abstract

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.

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,General Mathematics

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