Some bounded results of \(\theta (t)\)-type singular integral operators
type: Item
instance of: scholarly article
title: Some bounded results of \(\theta (t)\)-type singular integral operators
zbMATH Open document ID: 0963.47035
DOI: 10.1007/BF02459039
author: Kai Zhao
published in: Applied Mathematics and Mechanics. (English Edition)
publication date: 2000-01-01T00:00:00Z
review text: Let \(B\) be a Banach space in UMD (unconditionality of martingale differences) with an unconditional basis. Let \(L^p_B(\mathbb{R}^n)\) be the space of \(B\)-valued functions, \(1\leq p<\infty\) with the standard \(L^p_B\)-norm. A \(\theta(t)\)-type singular integral operator \(T: L^p_B(\mathbb{R}^n)\to L^p_B(\mathbb{R})\) is defined as \(T(f_j)= \text{p.v. }(k_j* f_j)= (\text{p.v. }k_j\neq f_j)\), where \(K= (k_j)\) satisfies the following conditions: a) \(\|K\|_\infty\leq C\); b) \(|K(x)|\leq C|x|^{-n}\) \((x\in \mathbb{R}^n\setminus \{0\})\), where \(\widehat K\) is the Fourier transform of \(K\) and \(C\) is a constant; c) \(|K(x)- K(x-y)|\leq C\theta({|y|\over|x|})|x|^{-n}\), \(2|y|<|x|\), where \(\theta(t)\) is a nondecreasing function on \([0,+\infty)\) with \(\theta(0)= 0\), \(\theta(2t)\leq C\theta(t)\) and \(\int^1_0{\theta(t)\over t} dt< +\infty\). Main results: \(T\) is weakly bounded in \(L^1_B(\mathbb{R}^n)\) and bounded on \(L^p_B(\mathbb{R}^n)\), \(1< p< +\infty\). \(T\) is bounded from the \(B\)-valued Hardy space \(H^1_B(\mathbb{R}^n)\) to \(L^1_B(\mathbb{R}^n)\) and bounded on \(H^1_B(\mathbb{R}^n)\).
Mathematics Subject Classification ID: 47G10
Mathematics Subject Classification ID: 45H05
zbMATH DE Number: 1506299
zbMATH Keywords: unconditionality
zbMATH Keywords: martingale differences
zbMATH Keywords: unconditional basis
zbMATH Keywords: \(\theta(t)\)-type singular integral operator
zbMATH Keywords: weakly bounded
zbMATH Keywords: \(B\)-valued Hardy space
reviewed by: Danuta Przeworska-Rolewicz
MaRDI profile type: Publication
cites work: Extension of a result of Benedek, Calderón and Panzone
cites work: Q4368156
cites work: Q4367338
cites work: Fourier series and Hilbert transforms with values in UMD Banach spaces
cites work: Q3803618
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description: scientific article; zbMATH DE number 1506299