\(L^ p\)-estimates for operators associated to oscillating plane curves
Let \(\Gamma:[-1,1]\to\mathbb{R}^ n\) be a continuous curve with \(\Gamma(0)=0\). Define the maximal function \(M_ \Gamma\) and the Hilbert transform \(H_ \Gamma\) by \[ M_ \Gamma f(z)=\text{sup}_{0<r\leq 1}{1\over r}\int^ r_ 0| f(z-\Gamma(t))| dt,\quad H_ \Gamma f(z)=\text{p.v. }{1\over r}\int^ 1_{-1}| f(z- \Gamma(t))|{dt\over t}, \] respectively. For curves of type \(\Gamma(t)=| t|^ \alpha\sin(| t|^{-\beta})\), \(\alpha,\beta>0\), and others which oscillate near the origin, the author proves that \(M_ \Gamma\) and \(H_ \Gamma\) are in \(L^ p,1<p\leq\infty\). This extends results of \textit{A. Carbery}, \textit{M. Christ}, \textit{J. Vance}, \textit{S. Wainger}, and \textit{D. K. Watson} [Duke Math. J. 59, No. 3, 675-700 (1989; Zbl 0723.44006)].
scientific article; zbMATH DE number 78554