Hilbert transforms and maximal functions along rough flat curves
Let \(\Gamma: \mathbb{R}\to \mathbb{R}^ n\) be a continuous curve of \(C'(\mathbb{R}- \{0\})\) with \(\Gamma(0)= 0\). Define \[ H_ \Gamma f(x)= \text{p.v. }\int^ \infty_{-\infty} f(x- \Gamma(t)) {dt\over t},\quad M_ \Gamma f(x)= \sup_{r>0} \int^ 1_ 0 | f(x- \Gamma(t))| dt. \] When \(\Gamma\) is well-curved at the origin, or when \(n= 2\) and \(\Gamma\) is flat and convex, there have been a series of positive results about the \(L^ p\)-boundedness of \(H_ \Gamma\) and \(M_ \Gamma\). In higher dimensional cases, which basic classes of curves should be studied was not clear enough. The paper introduces a simple class of curves and obtains an \(L^ p\)-boundedness result of \(H_ \Gamma\) and \(M_ \Gamma\) along such curves. The main result of the paper is: Let \(\Gamma(t)= (\gamma_ 1(t),\dots, \gamma_ n(t))\) be a balanced curve of class \((K)\), satisfying \[ \left|{\gamma_{j+1}'(\lambda t)\over \gamma_ j'(\lambda t)}\right|\geq 2\left|{\gamma_{j+1}'(t)\over \gamma_ j'(t)}\right|,\quad 1\leq j\leq n-1,\quad \lambda> 1,\quad t\neq 0. \] Then \(H_ \Gamma\) and \(M_ \Gamma\) are \(L^ p\)-bounded, \(1< p<\infty\). A curve is said to be in the class \((K)\) if \(\Gamma\) is continuous, \(\Gamma\in C'(\mathbb{R}- \{0\})\), \(\Gamma(0)= 0\), and \(|\gamma_ 1'|\), \(|{\gamma_{j+1}'\over \gamma_ j'}|\), \(j= 1,\dots, n-1\), are increasing for \(t>0\), and decreasing for \(t< 0\). \(\Gamma\) is said to be balanced if for some \(C\geq 1\), \(1\leq j\leq n\), and all \(t> 0\), we have \[ \left|{\gamma_ j(-t)\over \gamma_ j(ct)}\right|\leq 1,\quad \left|{\gamma_ j(t)\over \gamma_ j(- ct)}\right|\leq 1. \] {}.
scientific article; zbMATH DE number 647732