Oscillatory singular integral operators with variable Calderón-Zygmund kernels
type: Item
instance of: scholarly article
title: Oscillatory singular integral operators with variable Calderón-Zygmund kernels
zbMATH Open document ID: 0957.42008
author: Zusheng Zhou
author: Shanzhen Lu
author: Dachun Yang
published in: Southeast Asian Bulletin of Mathematics
publication date: 2001-03-07T00:00:00Z
review text: The authors consider the variable oscillatory singular operator defined by \[ Kf(x)= \text{p.v. }\int e^{iP(x,y)}k(x,x-y)f(y) dy, \] where \(P(x,y)\) is a real polynomial and \(k(x,y)\) is Caldéron-Zygmund kernel in \(y\) which satisfies regularity conditions uniformly in \(x\) for a.e. \(x\). Then the operator and corresponding maximal operator are bounded on \(L^p\) for \(1<p<\infty\), and the norm depends only on the degree of the polynomial \(P(x,y)\), but not on the coefficients. The commutators in the Coifman-Rochberg-Weiss sense \[ [a,K]f(x)= \text{p.v. }\int e^{iP(x,y)}k(x,x-y)(a(x)-a(y))f(y)d y \] are also considered. For \(a\in \text{BMO}\) the commutator appears to be bounded on \(L^p\) for \(1<p<\infty\). This follows from a more general result about the variable oscillatory singular operators of order \(N\). The proofs use ideas of \textit{F. Ricci} and \textit{E. M. Stein} [J. Funct. Anal. 73, 179-194 (1987; Zbl 0622.42010)].
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 47B47
zbMATH DE Number: 1406418
zbMATH Keywords: oscillatory singular integral operators
zbMATH Keywords: variable Caldéron-Zygmund kernel
zbMATH Keywords: bounded operators
zbMATH Keywords: \(L^p\)-spaces
zbMATH Keywords: maximal operator
zbMATH Keywords: commutator
reviewed by: Maria Roginskaya
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1406418