\(L^p\) boundedness of Carleson type maximal operators with nonsmooth kernels
type: Item
instance of: scholarly article
title: \(L^p\) boundedness of Carleson type maximal operators with nonsmooth kernels
zbMATH Open document ID: 1226.42008
published in: Tôhoku Mathematical Journal. Second Series
publication date: 2011-10-10T00:00:00Z
review text: The paper under review deals with the question of the \(L^p\) boundedness of a class of Carleson type maximal operators with rough kernels, i.e., ``standard'' kernels with no a priori smoothness assumptions. Namely, let \(\Omega\) be a measurable function on \(\mathbb R^n\backslash \{0\}\), homogeneous of degree 0 with zero average on \(S^{n-1}\), viz. \[ \Omega(tx)=\Omega(x) \quad \text{for any} \quad x \in \mathbb R^n\backslash \{0\} \quad \text{and} \;t>0, \] \(\Omega \in L^1(S^{n-1})\) and \[ \int_{S^{n-1}} \Omega(x') \,d\sigma (x')=0. \] Let \(Q_\lambda(r)=\sum_{2\leq k\leq d} \lambda_k r^k\), \(\lambda=(\lambda_2,\dots, \lambda_d)\in \mathbb R^{d-1}\). Then the Carleson type maximal operator \({\mathcal T}^*\) associated to the real-valued polynomial \(Q_\lambda(r)\) is defined by \[ {\mathcal T}^*(f)(x)=\sup_{\lambda} |T_\lambda (f)(x)|, \] where \(T_\lambda\) is an oscillatory integral given by \[ T_\lambda(f)(x)=\int_{\mathbb R^n} e^{iQ_\lambda(|y|)}K(y)f(x-y)\,dy, \] with \(K(y)=\Omega(y) / |y|^n\). The authors show that the mapping \(f\mapsto{\mathcal T}^*(f)\) is bounded in \(L^p(\mathbb R^n)\), \(1<p<\infty\), if \(\Omega \in H^1(S^{n-1})\), where \(H^1(S^{n-1})\) is the Hardy space on \(S^{n-1}\). The lengthy proof is based on an idea of linearizing maximal operators, Stein-Wainger's \(TT^*\) method and the Calderón-Zygmund rotation method (where the rotation method was used to overcome the lack of smoothness of the kernel). This result is, in the sense of removing the smoothness assumption on the kernel, an improvement on a result from \textit{E. M. Stein} and \textit{S. Wainger} [Math. Res. Lett. 8, No. 5--6, 789--800 (2001; Zbl 0998.42007)].
reviewed by: Humberto Rafeiro
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 42B25
zbMATH DE Number: 5955298
zbMATH Keywords: Carleson maximal operators
zbMATH Keywords: singular integral operators
zbMATH Keywords: rough kernel
zbMATH Keywords: oscillatory integrals
author: Yong Ding
author: Honghai Liu
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1309952088
OpenAlex ID: W1970221555
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DOI: 10.2748/TMJ/1309952088
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description: scientific article; zbMATH DE number 5955298