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Convex curves, Radon transforms and convolution operators defined by singular measures

Authors: Fulvio Ricci and Giancarlo Travaglini
Journal: Proc. Amer. Math. Soc. 129 (2001), 1739-1744
MSC (2000): Primary 42B10
Published electronically: October 31, 2000
MathSciNet review: 1814105
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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*}\Vert Tf\Vert _{L^{3}(\mathbb{T}\times \mathbb{R}^{2})}\leq c\Vert f\Vert _{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 \Vert Tf \right \Vert _{L^{s}\left (\mathbb{T} ,L^{p^{\pr... ...c\left \Vert f\right \Vert _{L^{p}\left (\mathbb{R}^{2}\right)} \end{equation*} where $p$ and $p^{\prime }$ are conjugate exponents.

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Additional Information

Fulvio Ricci
Affiliation: Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

Giancarlo Travaglini
Affiliation: Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca, Via Bicocca degli Arcimboldi 8, 20126 Milano, Italy

Keywords: Convolution operators, singular measures, Radon transforms
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
Article copyright: © Copyright 2000 American Mathematical Society

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