## $L^{p}-L^{p’}$ 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.## References

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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
- MR Author ID: 294667
- ORCID: 0000-0002-9670-9051
- Email: brandolini@unibg.it
**Allan Greenleaf**- Affiliation: Department of Mathematics, University of Rochester, Rochester, New York 14627
- Email: allan@math.rochester.edu
**Giancarlo Travaglini**- Affiliation: Dipartimento di Statistica, Università di Milano-Bicocca, Edificio U7, Via Bicocca degli Arcimboldi 8, 20126 Milano, Italy
- MR Author ID: 199040
- ORCID: 0000-0002-7405-0233
- Email: giancarlo.travaglini@unimib.it
- 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.
- © Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**359**(2007), 2559-2575 - MSC (2000): Primary 42B10, 44A12
- DOI: https://doi.org/10.1090/S0002-9947-07-03953-0
- MathSciNet review: 2286045