Convex curves, Radon transforms and convolution operators defined by singular measures
HTML articles powered by AMS MathViewer
- by Fulvio Ricci and Giancarlo Travaglini PDF
- Proc. Amer. Math. Soc. 129 (2001), 1739-1744 Request permission
Abstract:
Let $\Gamma$ be a convex curve in the plane and let $\mu \in M(\mathbb {R}^{2})$ be the arc-length measure of $\Gamma .$ Let us rotate $\Gamma$ by an angle $\theta$ and let $\mu _{\theta }$ be the corresponding measure. Let $T f(x,\theta )=f*\mu _{\theta }(x)$. Then \begin{equation*}\|Tf\|_{L^{3}(\mathbb {T}\times \mathbb {R}^{2})}\leq c\|f\| _{L^{3/2}(\mathbb {R}^{2})}. \end{equation*} This is optimal for an arbitrary $\Gamma$. Depending on the curvature of $\Gamma$, this estimate can be improved by introducing mixed-norm estimates of the form \begin{equation*}\left \| Tf \right \| _{L^{s}\left (\mathbb {T} ,L^{p^{\prime }}\left (\mathbb {R}^{2}\right )\right )}\leq c\left \| f\right \| _{L^{p}\left (\mathbb {R}^{2}\right )} \end{equation*} where $p$ and $p^{\prime }$ are conjugate exponents.References
- L. Brandolini, A. Iosevich and G. Travaglini, Spherical means and the restriction phenomenon, preprint.
- Luca Brandolini, Marco Rigoli, and Giancarlo Travaglini, Average decay of Fourier transforms and geometry of convex sets, Rev. Mat. Iberoamericana 14 (1998), no. 3, 519–560. MR 1681584, DOI 10.4171/RMI/244
- M. Christ, Endpoint bounds for singular fractional integral operators, unpublished.
- I. M. Gel′fand and G. E. Shilov, Generalized functions. Vol. 1, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1964 [1977]. Properties and operations; Translated from the Russian by Eugene Saletan. MR 0435831
- Daniel M. Oberlin, Multilinear proofs for two theorems on circular averages, Colloq. Math. 63 (1992), no. 2, 187–190. MR 1180631, DOI 10.4064/cm-63-2-187-190
- D. M. Oberlin and E. M. Stein, Mapping properties of the Radon transform, Indiana Univ. Math. J. 31 (1982), no. 5, 641–650. MR 667786, DOI 10.1512/iumj.1982.31.31046
- A. N. Podkorytov, On the asymptotics of the Fourier transform on a convex curve, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. vyp. 2 (1991), 50–57, 125 (Russian, with English summary); English transl., Vestnik Leningrad Univ. Math. 24 (1991), no. 2, 57–65. MR 1166380
- Burton Randol, A lattice-point problem, Trans. Amer. Math. Soc. 121 (1966), 257–268. MR 201407, DOI 10.1090/S0002-9947-1966-0201407-2
- Fulvio Ricci, $L^p$-$L^q$ boundedness for convolution operators defined by singular measures in $\mathbf R^n$, Boll. Un. Mat. Ital. A (7) 11 (1997), no. 2, 237–252 (Italian). MR 1477777
- Fulvio Ricci and Elias M. Stein, Harmonic analysis on nilpotent groups and singular integrals. III. Fractional integration along manifolds, J. Funct. Anal. 86 (1989), no. 2, 360–389. MR 1021141, DOI 10.1016/0022-1236(89)90057-8
- Robert S. Strichartz, Convolutions with kernels having singularities on a sphere, Trans. Amer. Math. Soc. 148 (1970), 461–471. MR 256219, DOI 10.1090/S0002-9947-1970-0256219-1
Additional Information
- Fulvio Ricci
- Affiliation: Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
- MR Author ID: 193872
- ORCID: 0000-0001-6272-8548
- Email: fricci@polito.it
- Giancarlo Travaglini
- Affiliation: Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca, Via Bicocca degli Arcimboldi 8, 20126 Milano, Italy
- MR Author ID: 199040
- ORCID: 0000-0002-7405-0233
- Email: travaglini@matapp.unimib.it
- Received by editor(s): May 15, 1999
- Received by editor(s) in revised form: September 27, 1999
- Published electronically: October 31, 2000
- Communicated by: Christopher D. Sogge
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 129 (2001), 1739-1744
- MSC (2000): Primary 42B10
- DOI: https://doi.org/10.1090/S0002-9939-00-05751-8
- MathSciNet review: 1814105