Multiparameter singular integrals and maximal operators along flat surfaces
type: Item
instance of: scholarly article
title: Multiparameter singular integrals and maximal operators along flat surfaces
zbMATH Open document ID: 1160.42006
DOI: 10.4171/RMI/566
author: Yong-Kum Cho
author: Sunggeum Hong
author: Joonil Kim
author: Chan Woo Yang
published in: Revista Matemática Iberoamericana
publication date: 2009-02-19T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.rmi/1228834304
full work available at URL: https://eudml.org/doc/42406
review text: As an extension of Calderón-Zygmund theory, singular Radon transforms and maximal functions along the appropriate submanifolds have been intensively studied. \textit{M. Christ, A. Nagel, E. M. Stein} and \textit{S. Wainger} [Ann. Math. 150, 489--577 (1999; Zbl 0960.44001)] showed their \(L^p\) boundedness under some curvature (or ``finite type'') condition. An interesting problem is to establish the \(L^p\) theory for flat manifolds, which are in lack of the finite type condition. \textit{H. Carlsson, M. Christ, A. Córdoba, J. Duoandikoetxea, J. L. Rubio de Francia, J. Vance, S. Wainger} and \textit{D. Weinberger} [Bull. Am. Math. Soc. 14, 263--267 (1986; Zbl 0588.44007)] considered \(L^p\) estimates for maximal functions and Hilbert transforms along flat curves in \(\mathbb{R}^2\). \textit{A. Seeger, S. Wainger, J. Wright} and \textit{S. Ziesler} [J. Geom. Anal. 15, 477--498 (2005; Zbl 1088.42007)] considered singular integrals associated with higher dimensional flat submanifold of the form \(\{ (t, \gamma ( | t | )) : t \in\mathbb{R}^n \}\). \textit{A. Carbery, S. Wainger} and \textit{J. Wright} [Duke Math. J. 101, 499--513 (2000; Zbl 0959.42006)] have found the necessary and sufficient condition for the \(L^p(\mathbb{R}^3)\) boundedness of the double Hilbert transforms associated with polynomial surfaces. The authors consider the \(L^p\) boundedness of the double Hilbert transforms along flat surfaces without finite type conditions. They consider model surfaces of the form \(\{ (t, \Gamma ( t )) : t \in \mathbb{R}^2 \}\) where \(\Gamma (t_1, t_2) = \gamma_1(t_1) \gamma_2(t_2)\). The double Hilbert transform along this surface is defined by \[ Hf(x) = \iint_{\mathbb{R}^1 \times \mathbb{R}^1} f( x_1 - t_1 , x_2 - t_2, x_3 - \gamma_1(t_1) \gamma_2(t_2)) \frac{dt_1}{t_1}\frac{dt_2}{t_2}. \] They prove the following. Suppose that \(\gamma_i\) is a convex function on \([0, \infty)\) such that \(\gamma_i(0) = \gamma'_i(0) =0\) for each \(i=1,2\). If each \(\gamma_i\) is…
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 42-02
zbMATH DE Number: 5509273
zbMATH Keywords: singular Radon transform
zbMATH Keywords: multiple Hilbert transform
zbMATH Keywords: flat surface
reviewed by: Yasuo Komori-Furuya
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5509273