A note on the existence and boundedness of singular integrals
type: Item
instance of: scholarly article
title: A note on the existence and boundedness of singular integrals
zbMATH Open document ID: 1020.42006
DOI: 10.1016/S0022-247X(02)00237-8
author: Yongzhong Sun
author: Zhifei Zhang
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-02-11T00:00:00Z
review text: Let \(K\in C({\mathbb R}^n\setminus\{0\})\) satisfy \(\int_{\varepsilon<|x|<N}K(x)dx=0\), for all \(0<\varepsilon<N<\infty\); there exists a constant \(C_1>0\) such that \[ |k(x)|\leq C_1|x|^{-n} \] for all \(x\neq 0\); there exists two positive constants \(C_2\) and \(\alpha\in (0,1]\) such that \[ |K(x-y)-K(x)|\leq C_2|y|^\alpha |x|^{-n-\alpha},\quad \forall |x|>2|y|. \] Define the operator \(T\) as follows: \[ T(f)(x)=\lim_{\varepsilon\rightarrow 0}\int_{|x-y|>\varepsilon}K(x-y)f(y)dy \] for \(f\in L^p({\mathbb R}^n)\), \(1\leq p<\infty\). In this paper, the authors prove that the operator \(T\) defines a bounded operator from \(L^\infty\) to the space BMO.
Mathematics Subject Classification ID: 42B20
zbMATH DE Number: 1866570
zbMATH Keywords: BMO
zbMATH Keywords: singular integral operators
zbMATH Keywords: John-Nirenberg inequality
reviewed by: Der-Chen Chang
MaRDI profile type: Publication
cites work: On the existence of certain singular integrals
cites work: \(H^p\) spaces of several variables
cites work: On functions of bounded mean oscillation
cites work: Calderon-Zygmund operators, pseudo-differential operators and the Cauchy integral of Calderon
cites work: Q5607484
cites work: Q3756738
description: scientific article; zbMATH DE number 1866570