Multilinear paraproducts revisited
type: Item
instance of: scholarly article
title: Multilinear paraproducts revisited
zbMATH Open document ID: 1342.42018
DOI: 10.4171/RMI/847
author: Loukas Grafakos
author: Danqing He
author: N. J. Kalton
author: Mieczysław Mastyło
published in: Revista Matemática Iberoamericana
publication date: 2015-10-20T00:00:00Z
review text: This 8-page note provides a simple new proof of the \(L^{p_0}\times L^{p_1}\times\cdots\times L^{p_m}\to L^{p,\infty}\) boundedness (for \(p_i\in[1,\infty)\) and \(p=(\frac{1}{p_0}+\frac{1}{p_1}+\ldots+\frac{1}{p_m})^{-1}\in[\frac{1}{m+1},\infty)\)) of multilinear paraproducts \[ P^{(k)}_{m+1}\vec{f}=\sum_j \Delta_j^{(0)}(f_0)\cdots\Delta_j^{(k)}(f_k)S_j^{(k+1)}(f_{k+1})\cdots S_j^{(m)} (f_m) \] built from Littlewood-Paley resolution operators \(\Delta_j^{(i)}\) and their partial sums \(S_j^{(i)}\). If \(k\geq 1\), so that there are at least two resolution operators, one gets an easy pointwise bound by a product of Littlewood-Paley square functions and Hardy-Littlewood maximal functions of the various \(f_i\). The \(L^{p,\infty}\) norm of this product is then controlled by a weak-type Hölder inequality and the well-known \(L^{p_i}\to L^{p_i,\infty}\) bounds for the two classical operators. In the main case that \(k=0\), each \(S^{(i)}_j\) is written as a sum of a finite number \(r_i\) of resolution operators (reducing to the above easy case) and a residual \(\tilde S^{(i)}_j=S^{(i)}_{j-r_i}\), so that the \(j\)-th term in the residual paraproduct \(g:=\tilde P^{(0)}_{m+1}(\vec{f})\) has Fourier support in an annulus \(|\xi|\sim 2^j\). Thus the Littlewood-Paley square function of \(g\) is controlled by a similar product as in the easy case. But the \(L^{p,\infty}\) norm of \(g\) itself is bounded by the \(L^{p,\infty}\) norm of its square function, thanks to a lemma of the second author [J. Fourier Anal. Appl. 20, No. 5, 1083--1110 (2014; Zbl 1309.42027)], which concludes the proof.
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 6496475
zbMATH Keywords: paraproducts
zbMATH Keywords: multilinear operators
zbMATH Keywords: Littlewood-Paley resolution operators
zbMATH Keywords: Hardy-Littlewood maximal functions
reviewed by: Tuomas Hytönen
MaRDI profile type: Publication
cites work: Carleson measures, trees, extrapolation, and \(T(b)\) theorems
cites work: Bilinear paraproducts revisited
cites work: Q5747968
cites work: Uniform estimates for paraproducts and related multilinear multipliers
cites work: Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires
cites work: Q3941791
cites work: A boundedness criterion for generalized Calderón-Zygmund operators
cites work: Classical Fourier Analysis
full work available at URL: https://doi.org/10.4171/rmi/847
OpenAlex ID: W2305217002
Recommended article: Bilinear paraproducts revisited
Recommended article: Multi-parameter paraproducts
Recommended article: Paraproducts for bilinear multipliers associated with convex sets
Recommended article: Uniform estimates for paraproducts and related multilinear multipliers
Recommended article: The linear bound for Haar multiplier paraproducts
Recommended article: Paraproducts in one and several parameters
Recommended article: Multilinear multiplier theorems and applications
Recommended article: Paraproducts via \(H^{\infty}\)-functional calculus
Recommended article: On multilinear Littlewood-Paley operators
Recommended article: Q4224826
description: scientific article; zbMATH DE number 6496475