Singular integrals along submanifolds of finite type
type: Item
instance of: scholarly article
title: Singular integrals along submanifolds of finite type
zbMATH Open document ID: 0967.42009
author: Kanghui Guo
author: Dashan Fan
author: Yi Biao Pan
published in: Michigan Mathematical Journal
publication date: 2001-01-01T00:00:00Z
review text: Let \(n\in \mathbb N\), \(n\geq 2\), and \(K(y)=\Omega(y)/|y|^n\) be a Calderón-Zygmund kernel on \(\mathbb R^n\). Let \(B(0, 1)\) be the unit ball centered at the origin in \(\mathbb R^n\). For \(d\in \mathbb N\) and a \(C^\infty\) mapping \(\Phi:B(0,1)\to\mathbb R^d\), define the singular integral operator \(T_\Phi\) on \(\mathbb R^d\) by \(T_\Phi(x)=\text{p.v.}\int_{B(0,1)}f(x-\Phi(y))K(y) dy\). The authors' main theorem is as follows: Suppose (i) \(\Phi\) is of finite type at \(0\), i.e., for each unit vector \(\eta\in \mathbb R^d\), there is a multi-index \(\alpha\) with \(|\alpha|\geq 1\) so that \(\partial_x^\alpha[\Phi(x)\cdot \eta]|_{x=0}\neq 0\), and (ii) \(\Omega\in L^q(S^{n-1})\) for some \(q>1\). Then \(T_\Phi\) is a bounded operator from \(L^p(\mathbb R^d)\) to itself for \(1<p<\infty\). They also give the \(L^p\) boundedness for the corresponding maximal truncated singular integrals. Their result is an improvement of E.~M.~Stein's one; Stein proved his result under the assumptions (i) and \(\Omega\in C^1(S^{n-1})\).
Mathematics Subject Classification ID: 42B20
zbMATH DE Number: 1548259
zbMATH Keywords: singular integrals
zbMATH Keywords: Calderón-Zygmund kernel
zbMATH Keywords: oscillatory integrals
reviewed by: Kôzô Yabuta
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1030132087
OpenAlex ID: W2023547726
DOI: 10.1307/MMJ/1030132087
description: scientific article; zbMATH DE number 1548259