Bi-parameter paraproducts
type: Item
instance of: scholarly article
title: Bi-parameter paraproducts
zbMATH Open document ID: 1087.42016
DOI: 10.1007/BF02392566
author: Camil Muscalu
author: Jill Pipher
author: Christoph Thiele
author: Terence Tao
published in: Acta Mathematica
publication date: 2005-06-29T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0310367
full work available at URL: fulltext.htm
review text: Let \(m\) be a bounded symbol on \(\mathbb R^4\), smooth away from the origin and such that \[ | \partial^{\alpha_1}_{\xi_1}\partial^{\alpha_2}_{\xi_2}\partial^{\beta_1}_{\eta_1}\partial^{\beta_2}_{\eta_2} m(\xi,\eta)\leq C| (\xi_1,\eta_1)| ^{-\alpha_1-\beta_1}| (\xi_2,\eta_2)| ^{-\alpha_2-\beta_2}, \] and denote \[ T(f,g)(x):= \int m(\xi,\eta)\widehat f(\xi)g(\eta)e^{2\pi i x(\xi+\eta)}\, d\xi\, d\eta. \] The main result of the paper states that \(T:L^p\times L^q\to L^r\) as long as \(1<p,q\leq \infty\), \(0<r<\infty\) and \(1/r=1/p+1/q\). To prove this fact, first \(T\) is discretized to a bi-parameter general paraproduct, and then a new technique in introduced to present a proof of the classical one-parameter case. This technique turns out to be very helpfull to handle an error term in the proof of the main theorem. The paper also contains two counterexamples concerning the double bilinear Hilbert transform \[ B(f,g):= \int f(x-t,y-s)g(x+t,y+s)(dt/t)(ds/s) \] and the trilinear operator \[ V(f,g,h)(x):= \int_{\xi_1<\xi_2<\xi_3} \widehat f(\xi_1)\widehat g(\xi_2)\widehat h(\xi_3) e^{2\pi i x(\xi_1-\xi_2+xi_3)} \, d\xi_1\, d\xi_2\, d\xi_3. \] The last part of the article contains some further comments and open questions.
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 47G10
Mathematics Subject Classification ID: 26D15
zbMATH DE Number: 2183720
zbMATH Keywords: paraproduct
zbMATH Keywords: bilinear operator
zbMATH Keywords: bilinear Hilbert transform
zbMATH Keywords: maximal function
reviewed by: Joan Cerdà
MaRDI profile type: Publication
arXiv ID: math/0310367
cites work: Carleson measures, trees, extrapolation, and \(T(b)\) theorems
cites work: Some recent developments in Fourier analysis and 𝐻^{𝑝}-theory on product domains
cites work: Polynomial growth estimates for multilinear singular integral operators
cites work: On the divergence of multiple Fourier series
cites work: Some Maximal Inequalities
cites work: A characterization of product BMO by commutators.
cites work: Multilinear Calderón-Zygmund theory
cites work: Calderón-Zygmund operators on product spaces
cites work: Commutator estimates and the euler and navier-stokes equations
cites work: Multilinear estimates and fractional integration
cites work: On Calderón's conjecture
cites work: Multi-linear operators given by singular multipliers
cites work: A counterexample to a multilinear endpoint question of Christ and Kiselev
cites work: \(L^p\) estimates for the biest. II: The Fourier case
cites work: Journé's covering lemma and its extension to higher dimensions
cites work: Q3142876
OpenAlex ID: W2081601543
description: scientific article