Averages in the plane over convex curves and maximal operators (Q580619)

Let \(\Gamma\) be the boundary of a compact convex centrally symmetric body in \({\mathbb{R}}^ 2\). Moreover, assume that \(\Gamma\) is sufficiently smooth and of non-vanishing curvature. Let \(\sigma\) denote the arc length measure of \(\Gamma\) and consider for \(t>0\) the averages \(A_ tf(x)=\int f(x+ty)\sigma (dy)\) and the corresponding maximal operator \(Mf=\sup_{t}| A_ tf|\). Then, the author proves the following Theorem: \(\| Mf\|_ p\leq C(\Gamma,p)\| f\|_ p,\) \(2<p<\infty\).