A T(1) theorem for entangled multilinear dyadic Calderón-Zygmund operators
type: Item
instance of: scholarly article
title: A T(1) theorem for entangled multilinear dyadic Calderón-Zygmund operators
zbMATH Open document ID: 1311.42040
author: Vjekoslav Kovač
author: Christoph Thiele
published in: Illinois Journal of Mathematics
publication date: 2014-11-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/1305.4752
full work available at URL: https://projecteuclid.org/euclid.ijm/1415023510
reviewed by: Pierre Portal
Mathematics Subject Classification ID: 42B20
zbMATH DE Number: 6369190
zbMATH Keywords: multilinear singular integral operators
zbMATH Keywords: T(1) theorem
zbMATH Keywords: Hilbert transform
zbMATH Keywords: paraproducts
MaRDI profile type: Publication
arXiv ID: 1305.4752
cites work: Carleson measures, trees, extrapolation, and \(T(b)\) theorems
cites work: Modulation invariant bilinear \(T(1)\) theorem
cites work: A boundedness criterion for generalized Calderón-Zygmund operators
cites work: On the two-dimensional bilinear Hilbert transform
cites work: Multilinear Calderón-Zygmund theory
cites work: Bellman function technique for multilinear estimates and an application to generalized paraproducts
cites work: Boundedness of the twisted paraproduct
review text: The paper under review deals with multilinear forms defined by NEWLINE\[NEWLINE \Lambda_{E}(F_{1},\dots,F_{|E|}) = \int \limits _{\mathbb{R}^{N}}K(x_{1},\dots,x_{N})\prod \limits _{(i,j) \in E} F_{(i,j)}(x_{i},x_{j})dx_{1}\dots dx_{N}, NEWLINE\]NEWLINE where \((\{1,\dots,N\},E)\) is a graph, and \(K\) is a kernel satisfying appropriate Calderón-Zygmund conditions. The case where \(N=2n\) and \(E=\{(i,n+i) \;;\; i=1,\dots,n\}\) corresponds to the influential multilinear theory developed by \textit{L. Grafakos} and \textit{R. H. Torres} [Adv. Math. 165, No. 1, 124--164 (2002; Zbl 1032.42020)]. Generalising this theory to more complicated graphs is important to study operators that are invariant under certain modulations. For instance, operators such as the two dimensional bilinear Hilbert transforms from [\textit{C. Demeter} and \textit{C. Thiele}, Am. J. Math. 132, No. 1, 201--256 (2010; Zbl 1206.42010)], correspond to the case where \(N=4\), \(E=\{(1,3), (1,4), (2,3)\}\) and \(K\) is of the form \((x_{1},...,x_{4}) \mapsto k(x_{1}-x_{2},x_{3}-x_{4})\). Obtaining \(L^p\) bounds for such operators is a fundamental question in harmonic analysis with important applications to ergodic theory. A key open question in this direction is whether or not the trilinear form associated with the triangular Hilbert transform NEWLINE\[NEWLINE \Lambda(F_{1},F_{2},F_{3}) = \int \limits _{\mathbb{R}^{3}}F_{1}(x,y)F_{2}(y,z)F_{3}(z,x)\frac{1}{x+y+z}dxdydz, NEWLINE\]NEWLINE is bounded on \(L^3\times L^3 \times L^3\). This corresponds to the case \(N=3\), \(E = \{(1,2),(2,3),(3,1)\}\).NEWLINENEWLINEThe paper under review is part of the development of a theory that aims to solve such open questions. At this stage, the triangular Hilbert transform remains out of reach, but simpler forms that can be thought of as building blocks for \(\Lambda\) can be treated. These forms arise from graphs \((\{1,\dots,N\},E)\) that have a bipartite structure: \(N=n+m\) and \(E \subset \{1,\dots,m\}\times\{1…
description: scientific article; zbMATH DE number 6369190