Boundedness criterion of Journé's class of singular integrals on multiparameter Hardy spaces
type: Item
instance of: scholarly article
title: Boundedness criterion of Journé's class of singular integrals on multiparameter Hardy spaces
zbMATH Open document ID: 1268.42024
author: Yongsheng Han
author: Guozhen Lu
author: Zhuo Ping Ruan
published in: Journal of Functional Analysis
publication date: 2013-03-11T00:00:00Z
review text: The authors study the boundedness of singular integral operators \(T\) with kernel \(\mathcal K\) in the Journé's class on \(\mathbb R^{n_1}\times \mathbb R^{n_2}\times \mathbb R^{n_3}\), with regularity exponent \(\varepsilon>0\) [\textit{J. L. Journé}, Rev. Mat. Iberoam. 1, No. 3, 55--91 (1985; Zbl 0634.42015)], acting on the multiparameter Hardy space \(H^p(\mathbb R^{n_1}\times \mathbb R^{n_2}\times \mathbb R^{n_3})\). Their main result reads as follows: For \(1\leq i\leq 3\), if \(T^*_i(1)=0\) denotes the condition that, for every \(\varphi^j\in C_{0,0}^\infty(\mathbb R^{n_j})\), \[ \int_{\mathbb R^{n_i}}\int_{\mathbb R^{n_1}\times \mathbb R^{n_2}\times \mathbb R^{n_3}} \mathcal{K}(x_1,x_2,x_3,y_1,y_2,y_3) \varphi^1(y_1)\varphi^2(y_2)\varphi^3(y_3)\,dy_1dy_2dy_3dx_i=0, \] then an operator \(T\) as above is bounded on \(H^p(\mathbb R^{n_1}\times \mathbb R^{n_2}\times \mathbb R^{n_3})\) for the range of indexes \(\max\{\frac{n_i}{n_i+\varepsilon}:1\leq i\leq 3\}<p\leq 1\), if and only if \(T^*_i(1)=0\), \(1\leq i\leq 3\). The necessary conditions are obtained from the cancellation properties of functions in the classical Hardy space, together with its maximal characterization. To prove the sufficiency, they reduce the problem to showing the boundedness of \(\{T_{t_1,t_2,t_3}\}_{t_1,t_2,t_3>0}\) from \(H^p(\mathbb R^{n_1}\times \mathbb R^{n_2}\times \mathbb R^{n_3})\) to \(L^p_{\mathcal H}(\mathbb R^{n_1+n_2+n_3})\) by using \(H^p\)-atoms, where \(T_{t_1,t_2,t_3}(f)=\psi_{t_1,t_2,t_3}*(T(f))\), \(\psi_{t_1,t_2,t_3}(x_1,x_2,x_3)=\psi_{t_1}^1(x_1)\psi_{t_2}^2(x_2)\psi_{t_3}^3(x_3)\), \(\psi_{t_i}^i(x_i)=t_i^{-n_i}\psi^i(x_i/t_i)\), \(\psi^i\in C_{0,0}^\infty(\mathbb R^{n_i})\) with \[ \int_0^\infty|\widehat{\psi^i}(t\xi)|^2\,\frac{dt}t=1,\quad\text{for all }\xi\in\mathbb R^{n_i}\setminus \{0\}, \] \(L^p_{\mathcal H}\) is the \(\mathcal H\)-valued \(L^p\) space and \(\mathcal H\) is the Hilbert space defined by \[ \mathcal{H}=\bigg\{\{h_{t_1,t_2,t_3}\}_{t_1,t_2,t_3>0…
reviewed by: Javier Soria
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 42B30
zbMATH DE Number: 6143076
zbMATH Keywords: Journé's class of singular integrals
zbMATH Keywords: Calderón--Zygmund operator
zbMATH Keywords: multiparameter Hardy space
zbMATH Keywords: almost orthogonality estimate
zbMATH Keywords: atomic decomposition
MaRDI profile type: Publication
full work available at URL: j.jfa.2012.12.006
OpenAlex ID: W1987885374
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DOI: 10.1016/J.JFA.2012.12.006
description: scientific article; zbMATH DE number 6143076