Discrete Calderón's identity, atomic decomposition and boundedness criterion of operators on multiparameter Hardy spaces
type: Item
instance of: scholarly article
title: Discrete Calderón's identity, atomic decomposition and boundedness criterion of operators on multiparameter Hardy spaces
zbMATH Open document ID: 1193.42090
published in: The Journal of Geometric Analysis
publication date: 2010-06-29T00:00:00Z
review text: The authors consider discrete Calderón's identity, a new atomic decomposition and give an application for this atomic decomposition in the setting of product Hardy space. Suppose \(0<p\leq 1\) and \(1<q<\infty\). Let \(\psi^{(1)}(x_1)\), \(\psi^{(2)}(x_2)\) be Schwartz functions with support on the unit ball satisfying the conditions: for a fixed large integer \(M\) depending on \(p\), \(\sum_j|\hat{\psi}^{(1)}(2^{-j}\xi_1)|^2=1\) for all \(\xi_1\in \mathbb R^n\setminus \{0\}\), and \(\int_{\mathbb R^n}\psi^{(1)}(x_1)x^{\alpha}_1dx_1=0\) for all \(0\leq|\alpha|\leq 2M\); and \(\sum_k|\hat{\psi}^{(2)}(2^{-k}\xi_2)|^2=1\) for all \(\xi_2\in \mathbb R^n\setminus \{0\}\), and \(\int_{\mathbb R^m}\psi^{(2)}(x_2)x^{\beta}_2dx_2=0\) for all \(0\leq|\beta|\leq 2M\). Then for every function \(f\in H^p(\mathbb R^n\times\mathbb R^m)\cap L^q(\mathbb R^{n+m})\), there exists a function \(g\in H^p(\mathbb R^n\times\mathbb R^m)\cap L^q(\mathbb R^{n+m}) \) with \(\| f\|_q\sim \| g\|_q\) and \(\| f\|_{H^p}\sim \| g\|_{H^p}\) such that the following discrete Calderón's identity \[ f(x_1,x_2)=\sum_j\sum_k\sum_I\sum_J|I\times J|\psi_{j,k}(x_1-x_I,x_2-y_J)\psi_{j,k}\ast g(x_I,y_J) \] holds true and the above series converges in both the norms of \(L^q(\mathbb R^{n+m})\) and \(H^p(\mathbb R^n\times\mathbb R^m)\), where \(I\subseteq \mathbb R^n\) and \(J\subseteq\mathbb R^m \) are dyadic cubes, \(x_I, y_J\) are any fixed points in \(I\) and \(J\), respectively. By using the above discrete Calderón's identity the authors give a new atomic decomposition in terms of \((p,q)\)-atoms for the multiparameter product Hardy space. Let \(f\in H^p(\mathbb R^n\times\mathbb R^m)\cap L^q(\mathbb R^{n+m})\). Then there exists a sequence of \((p,q)\)-product atoms \(\{a_j\}\) of \(H^p(\mathbb R^n\times\mathbb R^m)\) and a sequence of scalars \(\{\lambda_j\}\) with \((\sum_j|\lambda_j|^p)^{\frac{1}{p}}\leq C\| f\|_{H^p}\) such that \(f=\sum_j\lambda_ja_j\), where the series converges to \(f\) in both the …
Mathematics Subject Classification ID: 42B30
Mathematics Subject Classification ID: 42B20
zbMATH DE Number: 5728277
zbMATH Keywords: boundedness
zbMATH Keywords: multiparameter Hardy spaces
zbMATH Keywords: Calderón-Zygmund operator
zbMATH Keywords: Calderón's identity
zbMATH Keywords: atomic decomposition
zbMATH Keywords: boundedness criterion of operators
author: Yongsheng Han
author: Guozhen Lu
author: Kai Zhao
reviewed by: Yu Liu
MaRDI profile type: Publication
full work available at URL: s12220-010-9123-6
OpenAlex ID: W2152562276
cites work: Boundedness of operators on Hardy spaces via atomic decompositions
cites work: Q4059710
cites work: A continuous version of duality of \(H^ 1\) with BMO on the bidisc
cites work: The Calderon-Zygmund Decomposition on Product Domains
cites work: Some recent developments in Fourier analysis and 𝐻^{𝑝}-theory on product domains
cites work: A real variable characterization of $H^{p}$
cites work: Extensions of Hardy spaces and their use in analysis
cites work: Harmonic analysis on spaces of homogeneous type. With a preface by Yves Meyer
cites work: Harmonic analysis on product spaces
cites work: Littlewood-Paley theory on spaces of homogeneous type and the classical function spaces
cites work: Boundedness of operators on Hardy spaces
cites work: Duality of multiparameter Hardy spaces Hp on spaces of homogeneous type
cites work: Calderón-Zygmund operators on product spaces
cites work: A characterization of $H^{p}(R^{n})$ in terms of atoms
cites work: On the $H^1$--$L^1$ boundedness of operators
cites work: Q3135829
cites work: Q4338833
cites work: Duality in spaces of finite linear combinations of atoms
cites work: Q3142876
DOI: 10.1007/S12220-010-9123-6
description: scientific article; zbMATH DE number 5728277