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

Brandolini, Luca; Greenleaf, Allan; Travaglini, Giancarlo

doi:10.1090/S0002-9947-07-03953-0

$L^{p}-L^{p’}$ estimates for overdetermined Radon transforms
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by Luca Brandolini, Allan Greenleaf and Giancarlo Travaglini PDF

Trans. Amer. Math. Soc. 359 (2007), 2559-2575 Request permission

Abstract:

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.

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