Maximal functions and Hilbert transforms along variable flat curves
type: Item
instance of: scholarly article
title: Maximal functions and Hilbert transforms along variable flat curves
zbMATH Open document ID: 0959.42007
author: Anthony Carbery
author: Sonsoles Pérez
published in: Mathematical Research Letters
publication date: 2000-08-13T00:00:00Z
review text: Let \(S(x,y)\) be a suitable real-valued function on \({\mathbb R}^2\) vanishing on the diagonal. For \(f\in C^\infty_0({\mathbb R}^2)\), define \[ {\mathcal M}f(x)=\sup_{0<h<\infty}\frac 1h\left|\int^h_0 f(x_1-t,x_2-S(x_1,x_1-t)) dt\right| \] and \[ {\mathcal H}f(x)=\text{ p. v.}\int^\infty_\infty f(x_1-t,x_2-S(x_1,x_1-t)) \frac {dt}t. \] The authors establish the \(L^p({\mathbb R}^2)\) boundedness for \({\mathcal M}\) with \(1<p\leq\infty\) and for \({\mathcal H}\) with \(1<p<\infty\), via the \(L^p({\mathbb R}^2)\) estimates for certain singular integral operators with oscillatory terms.
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 47A30
Mathematics Subject Classification ID: 47B38
zbMATH DE Number: 1489489
zbMATH Keywords: Hilbert transform
zbMATH Keywords: maximal function
zbMATH Keywords: variable curve
zbMATH Keywords: oscillatory singular integral
reviewed by: Dachun Yang
MaRDI profile type: Publication
full work available at URL: mrl.1999.v6.n2.a12
OpenAlex ID: W2314462739
DOI: 10.4310/MRL.1999.V6.N2.A12
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description: scientific article; zbMATH DE number 1489489