Hilbert transforms along curves. II: A flat case
The author investigates the boundedness of operators H and M defined by \[ Hf(x)=pv\int^{1}_{-1}f(x-\gamma (t))t^{-1}dt\quad (f\in L^ 1_{loc}({\mathbb{R}}^ n)), \] \[ Mf(x)=\sup_{0<r\leq 1}r^{- 1}\int^{r}_{-r}| f(x-\gamma (t))| dt\quad (f\in L^ 1_{loc}({\mathbb{R}}^ n)), \] where \(\gamma\) (t) is a function from \([- 1,1]\) to \({\mathbb{R}}^ n\). His method yields the boundedness of various cases, however, we here note only the simple case of his results: Let \(\psi \in C^ 2(0,1)\) satisfy \(\psi (0)=\psi '(0)=0\), \(\psi''(t)>0\) \((0<t<1)\). Take \(\gamma(t)\) of the form \[ \gamma (t)=(t,\psi (t))\quad (0\leq t\leq 1);\quad \gamma (t)=(t,\psi (-t))\quad or\quad (t,-\psi (- t))\quad (-1\leq t<0). \] Then H and M are bounded as operators from \(L^ p({\mathbb{R}}^ 2)\) to itself for all \(p>1\).
scientific article; zbMATH DE number 4109319