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

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

Authors:	Fulvio Ricci Giancarlo Travaglini

Affiliation:	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 be a convex curve in the plane and let $mu in M(mathbb{R}^{2})$ be the arc-length measure of Let us rotate by an angle and let $mu_{theta }$ be the corresponding measure. Let $T f(x,theta)=fmu _{theta }(x)$ . Then $begin{equation}Vert TfVert _{L^{3}(mathbb{T}times mathbb{R}^{2})}leq cVert fVert _{L^{3/2}(mathbb{R}^{2})}. end{equation}$ This is optimal for an arbitrary . Depending on the curvature of , this estimate can be improved by introducing mixed-norm estimates of the form $begin{equation}left Vert Tf right Vert _{L^{s}left (mathbb{T} ,L^{p^{pr... ...cleft Vert fright Vert _{L^{p}left (mathbb{R}^{2}right)} end{equation*}$ where and $p^{prime }$ are conjugate exponents.

Keywords:	Convolution operators singular measures Radon transforms

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