Stability of approximation under the action of singular integral operators
type: Item
instance of: scholarly article
title: Stability of approximation under the action of singular integral operators
zbMATH Open document ID: 1123.46018
publication date: 2007-06-08T00:00:00Z
review text: Let \(X=L^p(\mathbb R^n,H),\) \(1<p<\infty,\) or \(X= \text{Lip}_\alpha (\mathbb R^n,H),\) \(0<\alpha <1,\) where \(H\) is an Euclidean space, and let \(T\) be a singular integral operator, bounded on \(X\) and having certain additional properties. Let the function \(f\) be such that \(f, Tf \in L^1(\mathbb R^n,H).\) Using a generalized Calderón-Zygmund decomposition, the authors construct for every \(t>0\) a function \(f_t\) from the ball \(B_X(ct)\) of radius \(ct\), centered at zero in the space \(X\), such that \[ \begin{aligned} \| f-f_t\| _{L^1} &\leq C \text{ dist}_{L^1} (f, B_X(t)), \\ \| Tf-Tf_t\| _{L^1} &\leq C(\| f-f_t\| _{L^1}+ \text{dist}_{L^1} (Tf, B_X(t))). \end{aligned} \]
Mathematics Subject Classification ID: 46B70
Mathematics Subject Classification ID: 42B20
zbMATH DE Number: 5163306
zbMATH Keywords: Calderón-Zygmund decomposition
zbMATH Keywords: singular integral operator
zbMATH Keywords: covering theorem
zbMATH Keywords: wavelets
author: S. V. Kislyakov
author: Natan Kruglyak
reviewed by: Georgi E. Karadzhov
MaRDI profile type: Publication
full work available at URL: s10688-006-0045-9
OpenAlex ID: W1998043450
cites work: Interpolation between \(H^ p\) spaces and non-commutative generalizations. I
cites work: Interpolation between \(H^ p\) spaces and non-commutative generalizations. II
cites work: Interpolation of Weighted and Vector-Valued Hardy Spaces
cites work: Some consequences of Pisier's approach to interpolation
cites work: Q3127985
cites work: Q4487423
cites work: Uniform algebras as Banach spaces
cites work: On the existence of certain singular integrals
cites work: Q3127986
cites work: Q5607484
cites work: Bounded orthogonal systems and the \(\Lambda\) (p)-set problem
cites work: Ten Lectures on Wavelets
cites work: Q5732767
DOI: 10.1007/S10688-006-0045-9
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 5163306