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

Authors: Luca Brandolini, Allan Greenleaf and Giancarlo Travaglini
Journal: Trans. Amer. Math. Soc. 359 (2007), 2559-2575
MSC (2000): Primary 42B10, 44A12
Published electronically: January 19, 2007
MathSciNet review: 2286045
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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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Additional Information

Luca Brandolini
Affiliation: Dipartimento di Ingegneria Gestionale e dell’Informazione, Università degli Studi di Bergamo, V.le G Marconi 5, 24044 Dalmine, Italy

Allan Greenleaf
Affiliation: Department of Mathematics, University of Rochester, Rochester, New York 14627

Giancarlo Travaglini
Affiliation: Dipartimento di Statistica, Università di Milano-Bicocca, Edificio U7, Via Bicocca degli Arcimboldi 8, 20126 Milano, Italy

Keywords: Radon transform, averages over curves, $L^{p}$ improving
Received by editor(s): December 16, 2003
Received by editor(s) in revised form: February 7, 2005
Published electronically: January 19, 2007
Additional Notes: The second author was partially supported by a grant from the National Science Foundation.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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