Hilbert transforms along curves. II: A flat case
type: Item
instance of: scholarly article
title: Hilbert transforms along curves. II: A flat case
zbMATH Open document ID: 0677.42011
DOI: 10.1215/S0012-7094-85-05246-9
author: Michael Christ
published in: Duke Mathematical Journal
publication date: 1985-01-01T00:00:00Z
review text: 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 \(p>1\).
reviewed by: Takafumi Murai
Mathematics Subject Classification ID: 42B20
zbMATH DE Number: 4109319
zbMATH Keywords: \(L^ p\) boundedness
zbMATH Keywords: Hilbert transform
zbMATH Keywords: maximal operator
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4109319