Paraproducts and the exponential-square class
type: Item
instance of: scholarly article
title: Paraproducts and the exponential-square class
zbMATH Open document ID: 1019.42010
DOI: 10.1016/S0022-247X(02)00121-X
author: J. Michael Wilson
published in: Journal of Mathematical Analysis and Applications
publication date: 2002-11-28T00:00:00Z
review text: Let \(b\) be a complex-valued bounded weight on \({\mathbb R}\) such that \(|b_D|>\delta\) for all dyadic intervals \(D\) (here \(b_D\) denotes \(b\)'s average over \(I\)), and let \(h_{(D),b}\) be the associated Haar functions introduced by \textit{R. R. Coifman, P. W. Jones} and \textit{S. Semmes} [J. Am. Math. Soc. 2, 553-564 (1989; Zbl 0713.42010)]. The main theorem in the paper under review shows that, if \(f\in L^1_{\text{loc}}\) has a pointwise bounded dyadic square function \[ S_{b}(f)(x)=\left(\sum_D \frac{|\lambda_{D,b}(f)|^2}{|D|}\chi_{D}(x)\right)^{1/2} \] associated to the Haar coefficients \(\lambda_{D,b}(f)\), then \(f\) is locally exponentially square integrable in the sense that \[ |\{ x; |f(x)-f_D|>\lambda\}|\leq c_1 |D|\exp(-c_2 \lambda^2) \] for all \(D\) and all \(\lambda >0\). This result is also extended to \(d\) dimensions for the natural Haar functions on \({\mathbb R}^d\).
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 42B25
zbMATH DE Number: 1836760
zbMATH Keywords: Littlewood-Paley functions
zbMATH Keywords: Haar functions
zbMATH Keywords: dyadic square function
zbMATH Keywords: paraproducts
reviewed by: Joan Cerdà
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1836760