Convex curves, Radon transforms and convolution operators defined by singular measures期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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

Authors:	Fulvio Ricci Giancarlo Travaglini

Institution:	Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy ; Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca, Via Bicocca degli Arcimboldi 8, 20126 Milano, Italy

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 $\begin{equation}\Vert Tf\Vert _{L^{3}(\mathbb{T}\times \mathbb{R}^{2})}\leq c\Vert f\Vert _{L^{3/2}(\mathbb{R}^{2})}. \end{equation}$ 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 and $p^{\prime }$ are conjugate exponents.

Keywords:	Convolution operators singular measures Radon transforms

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