On the \(T(1)\) theorem on product domains
type: Item
instance of: scholarly article
title: On the \(T(1)\) theorem on product domains
zbMATH Open document ID: 0804.42009
DOI: 10.1007/BF02835952
published in: Journal d'Analyse Mathématique
publication date: 1995-01-22T00:00:00Z
review text: The paper gives an \(\mathbb R^ n \times \mathbb R^ m\)-version of the Coifman-Meyer's theorem: Let \(T\) be a singular integral operator with its kernel \(K(x,y)\) satisfying a \(\delta\)-smoothness condition for \(y\), such that \(T(1) \in \text{BMO} (\mathbb R^ n)\), and \(T\) has the weak boundedness property. Then the operator \(S = \int^ \infty_ 0 q_ t Q_ t TP^ 2_ t {dt \over t}\) is \(L^ 2\)-bounded, where \(q_ t\), \(Q_ t\), and \(P_ t\) are the convolution operators with the nice functions \(\widetilde \psi_ t (x)\), \(\psi_ t (x)\), and \(\varphi_ t (x)\) \((\int_{\mathbb R^ n} \widetilde \psi (x)\, dx = 0 = \int_{\mathbb R^ n} \psi (x)\, dx\), \(\int_{\mathbb R^ n} \varphi (x)\, dx = 1)\), respectively. The result was the main step of the simpler proof of David-Journé's \(T(1)\) theorem. The product-version of this result asserts the \(L^ 2 (\mathbb R^ n) \times L^ 2 (\mathbb R^ m)\)-boundedness of the \(S'\) product-version. As its consequence, the paper gets some kind of \(T(1)\) theorem, which gives sufficient conditions for the boundedness of \(\delta\)-singular integral operator on \(\mathbb R^ n \times \mathbb R^ m\).
reviewed by: Ruilin Long
Mathematics Subject Classification ID: 42B20
zbMATH DE Number: 569433
zbMATH Keywords: product domain
zbMATH Keywords: David-Journé's \(T(1)\) theorem
zbMATH Keywords: Coifman-Meyer's theorem
zbMATH Keywords: singular integral operator
zbMATH Keywords: weak boundedness property
author: Josefina Alvarez Alonso
MaRDI profile type: Publication
cites work: Carleson measure on the bi-disc
cites work: A continuous version of duality of \(H^ 1\) with BMO on the bidisc
cites work: Q3941791
cites work: Q3739631
cites work: A boundedness criterion for generalized Calderón-Zygmund operators
cites work: \(H^p\) spaces of several variables
cites work: Calderón-Zygmund operators on product spaces
cites work: Q3719341
description: scientific article; zbMATH DE number 569433